Determinacy and Cardinal Invariants

Jindřich Zapletal ^{∗}

March 19, 2002

Abstract

Many standard forcing notions can be analyzed using determined infinite games. As a consequence, certain classes of independence results in set theory can be treated uniformly, based merely on the syntax of the sentences to be proved independent.

Contents

1 Introduction 3

2 Definable proper real forcings: general theory 4 2.1 The factor algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 A representation theorem . . . . . . . . . . . . . . . . . . . . . . 6 2.3 The resulting ideals . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Single step forcing: examples 15 3.1 Sacks forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Miller forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3 Laver forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.4 Cohen forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.5 Solovay forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.6 The E_{0} forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.7 The E_{1} forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.8 The c_{min} forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.9 Mathias forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 The countable support iteration 27 4.1 The geometric view of the iteration . . . . . . . . . . . . . . . . . 27 4.2 Transfinite Fubini powers of ideals . . . . . . . . . . . . . . . . . 32 4.3 The case of a projectively generated ideal . . . . . . . . . . . . . 34 4.4 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 5 Illfounded iterations 41

∗The author is partially supported by grants GA ČR 201-00-1466, NSF DMS-0071437 and a CLAS UF research award.

5.1 Strongly proper forcings . . . . . . . . . . . . . . . . . . . . . . . 41 5.2 Countable length . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 5.3 The uncountable length . . . . . . . . . . . . . . . . . . . . . . . 49 5.4 Sacks forcing iteration . . . . . . . . . . . . . . . . . . . . . . . . 50

6 Towers of ideals 55 6.1 Shooting a club with no infinite subset in the ground model … 57 6.2 Shooting a club with finite intersection with every ground model

ordertype ω set . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

7 Applications 64 7.1 Isolating cardinal invariants . . . . . . . . . . . . . . . . . . . . . 64 7.2 Optimality of the countable support iteration . . . . . . . . . . . 68 7.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 7.4 Interpolation theorems . . . . . . . . . . . . . . . . . . . . . . . . 71

A Some cardinal characteristics of the continuum 76

B The syntax of cardinal characteristics of the continuum 77 B.1 The covering numbers . . . . . . . . . . . . . . . . . . . . . . . . 77 B.2 The tame invariants . . . . . . . . . . . . . . . . . . . . . . . . . 80

C Large cardinals 81 C.1 The absoluteness results . . . . . . . . . . . . . . . . . . . . . . . 82 C.2 The determinacy results . . . . . . . . . . . . . . . . . . . . . . . 83

1 Introduction

This paper is an exposition of a certain connection between the large cardinal theory and forcing methodology that was discovered in 2000. This connection makes it possible in many interesting cases to indicate a canonical path towards a consistency result by merely considering the syntax of the sentence to be proved consistent. In many cases it also greatly simplifies the machinery of iterated forcing. Many applications of the method do not mention forcing at all and mention large cardinals in a way that is very unlikely to disturb the prospective users.

The prospective reader of this paper is a specialist in set theory. The reader is assumed to be reasonably proficient in forcing and in basic determinacy arguments as found in [7]. An encyclopedic knowledge of the cardinal invariants of the continuum is a good motivation, but not at all a necessary prerequisite. Bartoszynski’s book [31] is used as the generic reference. Similarly, no knowledge of the proofs of determinacy is assumed or necessary–the needed results are stated in Appendix C and can be easily used as black boxes in this paper as well as in all of its applications that I can imagine at this moment. Several of the necessary determinacy results remain unpublished at this time.

The paper is organized in the following way. Section 7 contains all the applications, and the reader should parse the statements of the theorems in this section before reading the other parts of the paper. Section 2 contains a characterization theorem for proper definable orderings adding a single real. Several examples of such forcings are closely studied in Section 3. Section 4 analyzes the countable support iteration of such forcings from the determinacy point of view. The subsequent two sections generalize the machinery to illfounded iterations and to construction of forcings adding interesting subsets of ω_{1}. The appendices contain the background information about some of the invariants used and the large cardinal black boxes.

The notation follows the set theoretic standard of [7]. The large cardinal assumptions are treated somewhat generously so as not to divert reader’s attention from the real issues. The use of such assumptions is denoted by a simple LC in the wording of the result; the optimalization of the hypotheses is left to the interested reader. A supercompact cardinal is always sufficient. The abbreviations AD and ADℝ denote the axioms of Determinacy and Real Determinacy respectively, asserting the determinacy of all games of length ω with integer or real moves. Given a Polish space X, the symbol B(X) denotes the algebra of all of its Borel subsets. Given a set or a class A and a tree on (A × ω)^{<ω} the projection of the tree T is the set p[T] = {r ∈ ω^{ω} : for some sequence s ∈ A^{ω} the pair s, r determines an infinite path through the tree T }. The Borel sets are constantly confused with their Borel definitions. Thus for a Borel set B and a forcing P the symbol Ḃ denotes the name for the Borel set in the P -extension with the same Borel definition as the set B. This usage will be extended to the universally Baire sets as well. The point is that the name Ḃ does not depend on the choice of the Borel (or universally Baire) definition of the set B. The “reals” are alternately realized as 2^{ω}, ω^{ω} etc.

2 Definable proper real forcings: general theory

The method described in this paper is motivated by the following basic fact: many posets adding a real are forcing-isomorphic to the ordering of the form P_{I} of all Borel sets of reals not in the ideal I ordered by inclusion, for a suitable σ-ideal I. The ordering P_{I} is not separative; its separative quotient is the factor σ-algebra B(ℝ)/I, but it is much easier to work directly with the ordering P_{I}. This section contains general remarks on the nature of such algebras in subsection 2.1, a representation theorem saying that every reasonably definable proper forcing adding a single real is forcing-equivalent to such an algebra in subsection 2.2, and finally a comment on the complexity of the resulting ideals in the last subsection.

2.1 The factor algebras

Lemma 2.1. Let I be a σ-ideal on the real line. The forcing extension by the poset P_{I} = B(ℝ) \ I is given by a single real; this is the unique real belonging to all sets in the generic filter.

Proof. For notational convenience put ℝ = ω^{ω}. Let ṙ be the P_{I}-name for a real defined by ṙ(ň) = m̌ if the set {s ∈ ℝ : s(n) = m} belongs to the generic filter. Note that this indeed defines a name for a total function from ω to ω since the collections {{s ∈ ℝ : s(n) = m} : m ∈ ω} are maximal antichains in the poset P_{I} for each integer n ∈ ω. Also note that if any real is to be in the intersection of all Borel sets in the generic filter, it must be ṙ. In order to complete the proof, it is enough to argue by induction on the complexity of the Borel set B ∉ I that B ⊩ ṙ ∈ Ḃ where Ḃ is the Borel set in the extension with the same Borel definition as B.

Now this is clearly true if B is a closed or a basic open set. Suppose B = ⋂_{n} A_{n} and for all n ∈ ω, A_{n} ⊩ ṙ ∈ Ȧ_{n} has been proved. Then for all n ∈ ω B ⊂ A_{n}, so B ⊩ ∀n ∈ ω ṙ ∈ Ȧ_{n} and B ⊩ ṙ ∈ ⋂_{n} Ȧ_{n} = Ḃ as desired. And suppose that B = ⋃_{n} A_{n} and for all n ∈ ω, A_{n} ⊩ ṙ ∈ Ȧ_{n} has been proved. The collection {A_{n} : A_{n} ∉ I} is a predense set in P_{I} below B–this simple observation uses the σ-completeness of the ideal I. So B ⊩ ∃n ∈ ω ṙ ∈ Ȧ_{n}, and B ⊩ ṙ ∈ ⋃_{n} Ȧ_{n} = Ḃ as desired. Since all Borel sets are obtained from closed sets and basic open sets by iterated applications of a countable intersection and union, the proof is complete.

The unique real from the statement of the previous lemma will be called the P_{I}-generic real and denoted by ṙ_{gen}. This real is forced to fall out of all ground model Borel sets in the ideal I. Another standard piece of terminology: if M is an elementary submodel of some large structure and r is a real such that the set {B ∈ P_{I} ∩ M : r ∈ B} is an M-generic filter on P_{I}, then I will call the real r M-generic.

The concept of an M-generic real is clearly important. Below I will use over and over the following basic complexity calculation.

Lemma 2.2. 1. Suppose that ⟨M, ∈⟩ is a countable model of set theory and P, ṙ ∈ M are a poset and a P -name for a real respectively. The set {r ∈ ℝ : for some M-generic filter G ⊂ P, r = ṙ/G} is Borel.

2. Let I be a σ-ideal on the reals. The poset P_{I} is proper if and only if for every Borel I-positive set B and every countable elementary submodel M of large enough structure containing B the set {r ∈ B : r is M-generic for P_{I}} is I-positive.

3. Let I be a σ-ideal such that P_{I} is proper. For every set B ∈ P_{I} and every P_{I}-name ẋ for a real there is a set C ∈ P_{I}, C ⊂ B and a Borel function f : C → ℝ such that C ⊩ ḟ(ṙ_{gen}) = ẋ. In particular, for every set B ∈ P_{I} and every universally Baire function g : B → ℝ there is a set C ∈ P_{I}, C ⊂ B such that g ↾ C is a Borel function.

Proof. Start with the first item. Within the model M, let Q be the complete Boolean subalgebra of the completion of the poset P generated by the name ṙ. The forcing theory says that X = {r ∈ ℝ : for some M-generic filter G ⊂ P, r = ṙ/G} = {r ∈ ℝ : for some M-generic filter G ⊂ Q, r = ṙ/G} and so it is enough to show that the latter set is Borel. To see that note that the set Y = {G ⊂ Q : G is an M-generic filter} is a Borel subset of P(Q): it is defined by the Borel conditions of “being a filter” and “meeting all of the countably many open dense sets in the model M”. Also, the set X is the image of Y under the continuous function G ↦ ṙ/G. Since the name ṙ generates the algebra Q, this function is one-to-one on Y. One-to-one continuous images of Borel sets are Borel, and so X is Borel.

For the second item, let I be a σ-ideal, B ∈ P_{I} a Borel set and M a countable elementary submodel of a large structure containing all the necessary information. Let C = {r ∈ B : r is M-generic}; the set C is Borel by (1). There are exactly two cases. Either C ∉ I; then C ∈ P_{I} is a condition in P_{I} which by Σ_{1}^{1} absoluteness forces the generic real to be M-generic, in other words it is an M-master condition. Or C ∈ I; in such a case B forces ṙ_{gen} ∉ Ċ, in other words that the generic real is not M-generic, so there cannot be an M-master condition below the set B. (2) follows.

For the third item, suppose that B ∈ P_{I} is a Borel set and ẋ is a name for a real. Let M be a countable elementary submodel of a large enough structure containing all the necessary information, and let C ⊂ B be the set of all Mgeneric reals in the set B. The function f : C → ℝ defined by f(r) = ẋ/r is Borel and by an absoluteness argument C ⊩ ḟ(ṙ_{gen}) = ẋ. The last sentence in (3) is just an application of this procedure to the name ẋ = ġ(r_{gen}) : for every real r ∈ C it must be the case that g(r) = s if and only if M[r] ⊧ g(r) = s by a wellfoundedness argument concerning the universally Baire trees used to define the function g.

An example of an application of the previous lemma:

Lemma 2.3. Suppose that I is a σ-ideal σ-generated by closed sets. The forcing P_{I} is proper.

Proof. Let B ∈ P_{I} be a positive Borel set and M be a countable elementary submodel of a large enough structure containing all the relevant information. By Lemma 2.2 it is enough to prove that the set C = {r ∈ B : r is M-generic} is positive. Suppose that F_{n} : n ∈ ω is a collection of closed sets in the ideal I. To show that C ⊄ ⋃_{n} F_{n}, fix an enumeration D_{n} : n ∈ ω of all open dense subsets of the poset P_{I} in the model M and by induction on n construct conditions B = B_{0} ≥ ⋯ ≥ B_{n} ≥ … in P_{I} ∩ M such that for each n ∈ ω B_{n+1} ∈ D_{n} and there is a basic open set O_{n} such that B_{n+1} ⊂ O_{n} and O_{n} ∩ F_{n} = 0. Once this is done, the unique real r ∈ ⋂_{n} B_{n} is in the set C and in none of the sets F_{n} as desired.

To perform the induction step, suppose the set B_{n} ∈ M ∩ P_{I} has been obtained. First of all, there must be a basic open set O_{n} such that B_{n} ∩ O_{n} ∉ I and O_{n} ∩ F_{n} = 0. If this were not the case then the set B_{n} could be written as the union of countable collection of I-small sets {F_{n} and B_{n} ∩ O : O a basic open set with empty intersection with F_{n}}. This is where the closure of the sets F_{n} is used. Look at the set B_{n} ∩ O_{n} ∈ M ∩ P_{I}. There must be a condition B_{n+1} ∈ D_{n} ∩ M below this set; this clearly concludes the inductive step.

The reader should observe that this argument really gives strong properness, and with a little bit of additional effort, < ω_{1}-properness [26] V.3.

The proof of the previous lemma does not resemble any previously known properness argument. It is the only proof of properness known to me whose hypotheses are purely syntactical. The ideals generated by closed sets include the ideal of countable sets (giving rise to Sacks forcing), the ideal of σ-bounded sets (for Miller forcing), the ideal of meager sets (for Cohen forcing) as well as some less well-researched ideals such as the ideal generated by closed measure zero sets. The associated forcings are studied in [41]. Many interesting proper posets (such as Laver forcing) are associated with ideals generated by more complex sets. No such a simple purely syntactical argument is available for them:

Example 2.4. Consider the ideal I σ-generated by the sets A_{r} = {s ∈ ℝ : s does not code r in some predetermined way of coding}, as r runs through all the reals. Clearly, the poset P_{I} collapses 𝔠 to ℵ_{0} since the generic real must code all the ground model reals. The ideal I is σ-generated by G_{δ} sets.

Another illustrative example of an ideal whose associated forcing is not proper is

Example 2.5. Let I be σ-generated by the sets A_{M} as M varies over all transitive countable models of set theory. The set A_{M} consists of all reals which are M-generic for some poset P_{J} ∈ M where M ⊧ J is a σ-ideal on the real line and P_{J} is proper.

2.2 A representation theorem

Let me now come back to the first sentence in this section. Which posets are forcing-isomorphic to the factor algebra P_{I} for a suitable σ-ideal I? By an old theorem of Sikorski [28], every c.c.c. poset adding a single real is. This must fail for proper forcings in general: the factor algebras have density ≤ 𝔠 and many proper posets adding a single real have higher density. However, a large important class of proper forcings can be represented by the factor algebras. It is not difficult to identify the relevant ideals:

Definition 2.6. Let P be a forcing adding a real ṙ_{gen}. I(P) is the ideal on the real line consisting of those sets A ⊂ ℝ for which Adam has a winning strategy in the game G(A). Eve starts, plays a condition p_{0} ∈ P and then the two players alternate to produce a descending chain p_{0} ≥ p_{1} ≥ p_{2} ≥ … in the poset P. Eve wins if the filter g generated by this chain meets all the antichains necessary for the valuation of the name ṙ_{gen}, and ṙ_{gen}/g ∈ A.

The definition depends on the name ṙ_{gen} for the generic real, but in all instances in this text the name will be clear from the context and I will suppress mentioning the dependence. On the other hand, the definition does not depend on the presentation of the poset P. The definition results in a quite well-known ideals, such as the Marczewski ideal. I explicitly state the definition in this way so that it is obvious that the ideal is obtained from the forcing and the name using a game quantifier. It is not difficult to see that

Lemma 2.7. I(P) is a σ-ideal.

Proof. The collection I(P) is clearly closed under subsets. Suppose now that A_{n}, n ∈ ω are sets in I(P) and σ_{n}, n ∈ ω are the relevant Adam’s strategies. I must produce a winning strategy for Adam in the game G(⋃_{n} A_{n}). Well, Adam will win if he creates the following log along with his moves:

p_{0} p_{1} p_{2} aux_{0} p_{4} aux_{1} p_{6} aux_{3} …
aux_{0} p_{3} aux_{1} aux_{2} aux_{3} aux_{4} …
aux_{2} p_{5} aux_{4} aux_{5} …
aux_{5} p_{7} …
...

Here the lines are simulated plays respecting the strategies σ_{0}, σ_{1}, … so Eve’s moves in these plays are recorded in columns 0, 2, 4 … The entries p_{0}, p_{1}, p_{2} … are the ones actually made in the run of the game G(⋃_{n} A_{n}) that Adam is trying to win while the aux entries are auxiliary. The only other rule for keeping the log is that each simulated Eve’s move is equal to the simulated Adam’s move just above and to the right from it.

There is clearly exactly one way of keeping the log. The columns 2m and 2m + 1 are always obtained together from top to bottom, the condition p_{2m} is the weakest in these two columns and p_{2m+1} is the strongest. After the game is over, the filter g generated by the descending chain of conditions actually played is equal to each of the filters g_{n} generated by the descending chain in the simulated play respecting the strategy σ_{n} in the n-th row of the log. Thus for each n ∈ ω, ṙ_{gen}/g = ṙ_{gen}/g_{n} ∉ A_{n}, so ṙ_{gen} ∉ ⋃_{n} A_{n} and Adam has won.

Now clearly without further assumptions on the poset P it cannot be expected that P is equivalent to P_{I(P)}. But for a large class of forcings this is indeed true as shown in the following definition and theorem.

Definition 2.8. ⟨P, ≤⟩ is a real forcing if the conditions in P are real numbers and there is a P -name ṙ_{gen} for a real and a projective formula χ(x, y) such that P ⊩the generic filter is the set {p ∈ P̌ : χ(p, ṙ_{gen})}. The poset is called a definable real forcing if P, ≤ and ṙ_{gen} are projective sets. For the name ṙ_{gen} this means that the set {(p, n, m) ⊂ ℝ × ω × ω : p ⊩ ṙ_{gen}(ň) = m̌} is a projective subset of ℝ × ω × ω.

Many forcings used in the theory of cardinal invariants fall into this syntactically identified class of partial orders [31], [20]. Note that in the presence of large cardinals guaranteeing absoluteness of the formula χ, the generic filter on P can be recovered in V [r_{gen}] as the set {p ∈ P̌ : χ(p, ṙ_{gen})}.

Theorem 2.9. (ZFC+LC) Every proper real forcing is forcing equivalent to the poset P_{I} where I = I(P).

The large cardinal hypothesis is necessary only if the formula χ is more complicated than analytic–a situation that has never occurred in practice. In fact one of the obvious consequences of the theorem is that under suitable large cardinal assumptions every real forcing can be represented so that the formula χ is analytic–the P_{I} is such a representation.

Proof. Fix the forcing P, the name ṙ_{gen} and the formula χ. The argument begins with a small observation. If B ⊂ ℝ is a Borel or even universally Baire Ipositive set and M is a countable elementary submodel of large enough structure containing all the necessary information including the set B then the set C ⊂ B of all M-generic reals for the poset P which are in the set B, is Borel and I-positive.

Now the set C is Borel by Lemma 2.2–note that C = {r ∈ ℝ : r is M-generic and M[r] ⊧ r ∈ B}. For the positivity, suppose that σ̄ is Adam’s strategy in the game G(C). Eve will defeat it in the following manner. By the elementarity of the model M and the positivity of the set B there is a condition p such that Eve has not lost the game G(B) after playing p in the first move. Let p_{0} ≤ p be an M-master condition. Clearly Eve is not lost in G(B) even if she plays p_{0} instead of p. Now consider the following Adam’s strategy σ in the game G(B) after that move. As the play p_{0} ≥ q_{0} ≥ p_{1} ≥ q_{1} ≥ … develops, Adam will keep a log of conditions q̄_{0}, q̄_{1}, … so that p_{n} ≥ q̄_{n} ≥ q_{n} and p_{0} ≥ q̄_{0} ≥ p_{1} ≥ q̄_{1} ≥ … is a play according to the strategy σ̄; moreover Adam will make sure that q_{n} has a condition in O_{n} ∩ M above it, where O_{n} is the n-th open dense subset of the poset P in the model M in some fixed enumeration. This Adam can certainly do, as p_{0} is an M-master condition. Since Eve was not lost after her first move, there must be a play against the strategy σ in which Eve wins, that is, writing g for the resulting filter on P, r = ṙ_{gen}/g ∈ B. Now the strategy σ was chosen so that the filter g is M-generic and there is a play against the strategy σ̄ generating the same filter. Clearly, Eve won that play in the sense of the game G(C) as desired.

Now it is possible to show that P_{I} ⊩ Ḣ = {p ∈ P̌ : χ(p, ṙ_{gen})} is a generic filter on P, where ṙ_{gen} denotes the canonical P_{I}-generic real. Suppose that B is a Borel I-positive set, p_{0} and p_{1} are conditions in P such that B ⊩ p̌_{0}, p̌_{1} ∈ Ḣ and let O ⊂ P be an open dense subset. I will produce a set D ⊂ B and a condition q ∈ P such that q ∈ O and q ≤ p_{0}, p_{1}, and D ⊩ q̌ ∈ Ḣ. This will prove that Ḣ is forced to be a generic filter.

Let M be a countable elementary submodel of a large enough structure containing all the relevant information and let C ⊂ B be the set of all Mgeneric reals for the poset P in the set B. As in the previous paragraph this set is I-positive and Borel, therefore an element of P_{I(P)}. By a projective forcing absoluteness argument C ⊩ Ḣ ∩ M̌ ⊂ P̌ ∩ M̌ is an M̌-generic filter containing the conditions p̌_{0}, p̌_{1}. So there must be a Borel I-positive set D ⊂ C and a condition q ∈ O ∩ M below both p_{0} and p_{1} such that D ⊩ q̌ ∈ Ḣ as desired.

To complete the proof of the theorem it is only necessary to show that for each condition p ∈ P there is a Borel I-positive set B such that B ⊩ p̌ ∈ Ḣ. Let C = {r ∈ ℝ : χ(p, r)}. Clearly, C is a projective set and if it is I-positive, any Borel I-positive subset of it obtained as in the first paragraph will force p̌ into Ḣ by a projective absoluteness argument. Now Eve will in fact win in the game G(C): as the play p = p_{0} ≥ p_{1} ≥ p_{2} ≥ … develops she will keep a log of countable elementary submodels M_{n}, n ∈ ω of some large enough structure making sure that M_{0} ∈ M_{1} ∈ …, p_{n} ∈ M_{n} and that in the end the filter g generated by the descending chain played is M = ⋃_{n} M_{n}-generic. Obviously p ∈ g, so M[g] ⊧ χ(p, ṙ_{gen}/g) and since the model M[g] is correct about projective truth, χ(p, ṙ_{gen}/g) holds. Thus ṙ_{gen}/g ∈ C and Eve won as desired.

2.3 The resulting ideals

The whole method presented in this text depends on the fact that there is a certain correlation between the forcing properties of the poset P and the descriptive set theoretic properties of the ideal I(P). I will consider a natural descriptive set theoretic analysis of this ideal. The Borel part of the ideal I(P) will be its subideal generated by its Borel members, and similarly for the universally Baire part. In general, given a Wadge pointclass Γ the Γ-part of the ideal I(P) will be the subideal generated by the collection Γ ∩ I(P). Clearly, only the Borel part of the ideal I(P) is important for the evaluation of the poset P_{I(P)}, and the ideal I(P) can be obtained from its Borel part in a very simple way:

Lemma 2.10. (ZFC+LC) Let P be a proper real forcing. A set A ⊂ ℝ is in the ideal I(P) if and only if every I(P)-positive Borel set has a positive Borel subset disjoint from A. In particular, a universally Baire set A ⊂ ℝ is in the ideal if and only if P ⊩ ṙ_{gen} ∉ Ȧ.

The perfect set theorem can be applied to show that the Borel part of the ideal I(Sacks) is the ideal of countable sets, and this claim then implies that the ideal I(Sacks) is the Marczewski ideal.

Proof. Look at the first equivalence. The right-to-left direction is almost trivial and left to the reader. For the left-to-right direction, suppose that A ⊂ ℝ is a set in the ideal I(P) and σ is a winning strategy for Adam in the game G(A). Let B ⊂ ℝ be a Borel positive set; I must produce a positive Borel subset of it disjoint from A.

First observe that P ⊩ there is a play of the game G(A) observing the strategy σ such that the moves of the play come from the generic filter and they meet all the dense subsets of P necessary to evaluate the name ṙ_{gen}. In the P -generic extension V [G], such a play is constructed by induction. If conditions p_{0} ≥ ⋯ ≥ p_{2n+1}, all in the filter G, have been constructed, then the set {q ∈ P̌ : ∃q′ ≤ p_{2n+1} σ(p_{0}, … p_{2n+1}, q′) = q} is dense below p_{2n+1}. By the genericity of the filter G there must be a condition p_{2n+3} in the intersection of the filter G and this set deciding the value of ṙ_{gen}(ň). The condition p_{2n+2} is found accordingly.

Now let M be a countable elementary submodel of a large enough structure containing all the relevant information and consider the set of all M-generic reals in the set B. This is an I(P)-positive Borel subset of B and by the previous paragraph it must be disjoint from the set A, since for each real in it there is a play of the game which follows the winning strategy σ and results in that very real.

The second equivalence now follows immediately from the universally Baire absoluteness C.4.

However, it turns out that the study of larger parts of the ideal I(P) does have its rewards. I will comment on two issues in this direction. The first is an attempt to find a formulation of the statement “P_{I} is proper” expressible in the language of the model L(ℝ)(Γ)[I], where Γ is some class of universally Baire sets, possibly empty.

Definition 2.11. An ideal I is ordinal covering if for every I-positive set B, every ordinal λ ∈ θ and every function f : B → [λ]^{ℵ_{0}} there is a countable set x ⊂ λ such that f^{−1}[x]^{ℵ_{0}} is I-positive. If Γ is a Wadge class of universally Baire sets, the ideal I is called Γ-ordinal covering if the above holds for every λ ∈ θ^{L(ℝ)(Γ)} and every B, f ∈ L(ℝ)(Γ).

Clearly, the statement “I is Γ-ordinal covering” is expressible in the model L(ℝ)(Γ)[I]. The following lemmas correlate this statement with the properness of the poset P_{I}. The first lemma shows that ordinal-covering ideals occur frequently in nature, in particular the ideals I(P) are Γ-covering if the poset P is proper. The second lemma shows that the ordinal covering properties of an ideal I have consequences reminiscent of the properness of the forcing P_{I}.

Lemma 2.12. Suppose that Γ is a class of universally Baire sets such that the model L(ℝ)(Γ) satisfies Real Determinacy and Dependent Choice and Γ = L(ℝ)(Γ) ∩ P(ℝ).

1. Suppose P is a proper real forcing. Then the ideal I(P) is Γ-ordinal covering.

2. Suppose that I is a projectively defined projectively generated ideal and there is a proper forcing Q introducing a set in the ideal İ containing all ground model I-small sets as subsets. Then the ideal I is Γ-covering.

3. Suppose J is an arbitrary σ-ideal. Then there is the smallest σ-ideal I ⊃ J which is Γ-ordinal covering.

Proof. For the first item, let B ∈ Γ be a positive set and f ∈ L(ℝ)(Γ), f : B → [λ]^{ℵ_{0}} be a function as in the definition of ordinal covering. Let r be a P -generic real, r ∈ B. By Fact C.5 there is an embedding j : L(ℝ)(Γ)^{V} → L(ℝ)(Γ)^{V [r]} fixing all ordinals. Thus jf(r) is a countable subset of j(λ) = λ and by the properness of the forcing P there must be a ground model countable set x ⊂ λ such that jf(r) ⊂ x. Now x ∈ L(ℝ)(Γ)^{V}, jf(r) ⊂ x = j(x) and so r ∈ j(f^{−1}[x]^{ℵ_{0}}). Thus x is the sought countable subset of λ with the preimage f^{−1}[x]^{ℵ_{0}} I-positive.

The proof of the second item is essentially the same, replacing the poset P with Q and choosing the real r ∈ B in the Q-extension so that it falls out of all ground model coded I-small sets of reals. This is possible since by C.4 the set Ḃ is I-positive in the Q-extension and therefore is not a subset of ⋃(I ∩ V) which is I-small in the extension by the assumptions.

For the third item, fix an arbitrary ideal J on the real line. To get the ideal I ⊃ J by induction on ordinal α define σ-ideals I_{α} so that

• I_{α+1} is the σ-ideal generated by I_{α} and all sets A ⊂ ℝ in Γ such that there is a function f : A → [λ]^{ℵ_{0}} in L(ℝ)(Γ) such that for every countable set x ⊂ λ the preimage f^{−1}[x]^{ℵ_{0}} is I_{α}-small.

• I_{α} is the σ-ideal generated by ⋃_{β∈α} I_{β} at limit ordinals α.

This construction is easily seen to produce an inclusion-increasing sequence of σ-ideals on ℝ. By Replacement, the construction has to stabilize at some ordinal α; let I be this stable value. The ideal I is clearly the smallest ordinalcovering ideal containing J.

Lemma 2.13. Let Γ be a pointclass of universally Baire sets such that Γ = L(ℝ)(Γ) ∩ P(ℝ) and the model L(ℝ)(Γ) satisfies the Real Determinacy and Dependent Choice. Let I be a Γ-covering σ-ideal. Then

1. the ideal I is closed under well-ordered unions from the model L(ℝ)(Γ).

2. Every I-positive set in Γ has a Borel I-positive subset.

3. For every relation R ⊂ ℝ × ℝ in the class Γ such that the set {r ∈ ℝ : ∃s ∈ ℝ ⟨r, s⟩ ∈ R} is I-positive, there are a Borel I-positive set B and a Borel function f : B → ℝ such that for every real r ∈ B it is the case that ⟨r, f(r)⟩ ∈ R.

Proof. For the first item suppose a set A ∈ Γ is a union ⋃_{α∈λ} B_{α} of I-small sets. Without loss of generality this is a union of disjoint sets and λ ∈ θ^{L(ℝ)(Γ)}. If A were a positive set then considering the function f : A → [λ]^{ℵ_{0}}, f(r) = {α} for the unique α ∈ λ such that r ∈ B_{α}, there would have to be a countable set x ⊂ λ such that f^{−1}[x]^{ℵ_{0}} = ⋃_{α∈x} B_{α} is positive, contradicting the assumption that I is a σ-ideal.

For the second item, let A be a positive set in Γ. Use the Real Determinacy C.9 to find a tree T ⊂ (λ × ω)^{<ω} in the model L(ℝ)(Γ) for some ordinal λ ∈ θ such that A = p[T]. Define f(r) for r ∈ A as the set of all ordinals occurring in the leftmost branch of the tree T producing the real r. Let x ⊂ λ be a countable set such that f^{−1}[x]^{ℵ_{0}} is positive. This set is a subset of the analytic set p[T ∩ (x × ω)^{<ω}] ⊂ A, which therefore must be positive as well. Every analytic set is an ω_{1}-union of Borel sets. The closure of the ideal under well-ordered unions then produces the desired positive Borel subset of the set A.

The proof of the third item is similar and left to the reader.

Now suppose that J is a projectively generated projectively defined ideal, and we want to increase cov(J) by a proper notion of forcing. The above lemmas suggest the following heuristic for doing it: choose a class Γ of universally Baire sets such that L(ℝ)(Γ) ∩ P(ℝ) = Γ and the model L(ℝ)(Γ) satisfies the Real Determinacy and Dependent Choice, and let I be the smallest Γ-ordinal covering σ-ideal larger than J. The poset P_{I} will then be a natural candidate for a proper forcing increasing cov(J). This heuristic sometimes works, most notably in Section 5, but it may fail, as Subsection 3.7 shows. It is unfortunately false that for every definable ideal J there would be the smallest larger σ-ideal I such that P_{I} is proper.

The second issue to be noted is the possibility that in fact the universally Baire part of the ideal I(P) is generated by the Borel part or the projective part of it as an ideal. This happens reasonably often and it has strong correlation with the forcing properties of the poset P:

Lemma 2.14. If the real forcing P is c.c.c. then the ideal I(P) is equal to its Borel part.

Proof. Let A ∈ I(P) be an arbitrary set. By Lemma 2.10 there is a dense set of Borel sets in the poset P_{I(P)} disjoint from the set A. Find a maximal antichain X in this dense set. Since the forcing is c.c.c., necessarily X is countable and ⋃ X is a Borel set such that ℝ \ ⋃ X ∈ I(P). Clearly A ⊂ ℝ \ ⋃ X.

Lemma 2.15. (ZFC+LC) Suppose that the definable real forcing P is provably < ω_{1}-proper. If the universally Baire part of the ideal I(P) is equal to its projective part then every intermediate subextension of the P -extension is c.c.c.

Proof. Identify the forcings P and P_{I(P)}. Without loss I can assume that the Continuum Hypothesis holds–if it does not, just force it without adding reals, not changing the poset P and apply the lemma in the extension. Let Q be a nowhere c.c.c. partial order regularly embeddable into P. Let Ḣ be a P -name for a Q-generic filter. I will show that P ⊩ ṙ_{gen} ∈ V [Ḣ].

If M is a countable elementary submodel of a large enough structure and r is a real which falls out of all I(P)-small sets in the model M, the set {B ∈ M ∩P : r ∈ B} is a filter on M ∩ P_{I(P)} and the set {q ∈ Q ∩ M : ∃B ∈ M B ⊩_{P} q̌ ∈ Ḣ} is a filter on M ∩ Q. The latter filter I will denote by Ḣ ∩ M/r. The key point is that if the poset Q is not c.c.c. this filter may or may not be M-generic.

Claim 2.16. Let B ∈ P be a Borel set and M_{0} ∈ M_{1} ∈ … an ω-tower of countable elementary submodels of large enough structure. For every infinite set a ⊂ ω the set B_{a} = {r ∈ B : n ∈ a ↔ Ḣ ∩ M_{n}/r is M_{n}-generic} is an I(P)-positive set.

Granted the claim, move into a model L(ℝ)(Γ) of Real Determinacy and Dependent Choice all of whose sets of reals are universally Baire. This model contains the ideal I(P) ∩ Γ, the collection {B_{a} : a ⊂ ω infinite} of I(P)-positive sets, and since the ideal I(P) ∩ Γ is projectively generated, it also contains a basis {A_{a} : a ⊂ ω infinite} for it indexed by reals. Use the Uniformization C.9 to find a function f : [ω]^{ω} → ℝ such that f(a) ∈ B_{a} \ C_{a}. The range of this function is I(P)-positive since it contains the point f(a) outside of every basis set C_{a}. As a universally Baire positive set, it has a positive Borel subset C ⊂ B. Move out of the model L(ℝ)(Γ). By an absoluteness argument between V and V^{P}, the set C forces that ṙ_{gen} is the only real number r ∈ Ċ such that Ḣ ∩M_{n}/r is M_{n}-generic if and only if Ḣ ∩ M̌_{n} is M_{n}-generic. By another absoluteness argument, this time between V^{Q} and V^{P}, the condition C then forces that ṙ_{gen} ∈ V [Ḣ] as desired.

The proof of Claim 2.16 is based on the following simple observation: since the forcing Q is nowhere c.c.c. of density 𝔠 = ℵ_{1}, a density argument shows that for every continuous ∈-tower N_{α} : α ∈ ω_{1} of countable elementary submodels of large enough structure the generic filter on Q will not be generic for all the models N_{α}. Now fix the positive Borel set B, the ∈-tower M_{n} : n ∈ ω and an infinite set a ⊂ ω. For each number n ∉ a choose a continuous ∈-tower ⟨N_{α}^{n} : α ∈ ω_{1}⟩ of countable elementary submodels of large enough structure in the model M_{n} in such a way that M_{n−1} ∈ N_{0}^{n}. Consider the countable continuous ∈-tower T_{a} consisting of the models M_{n} : n ∈ a and N_{α}^{n} : n ∉ a, α ≤ M_{n} ∩ ω_{1}. Since the forcing P is < ω_{1}-proper, below the set B there is a master condition for all the models on the tower T_{a} simultaneously, and so the set C_{a} = {r ∈ B : r is a generic real for all the models on the tower T_{a}} is I(P)-positive. The proof will be complete if I show that C_{a} ⊂ B_{a}. And indeed, if r ∈ C_{a} then clearly r is M_{n}-generic for all n ∈ a. On the other hand, for n ∉ a the real r cannot be M_{n}-generic because if it were, it would have to be not generic for one of the models N_{α}^{n} : α ∈ ω_{1} ∩ M_{n} by the elementarity of the model M_{n}, contradicting the definition of the set C_{a}.

There are several typical cases of the trivialization of the universally Baire part of the ideal I(P) mentioned in Lemma 2.15. The first case is that of c.c.c. forcings. Another case are forcings such as Sacks and Laver. There an integer game and a continuous function f : ω^{ω} → ℝ is associated with the forcing so that a universally Baire set A is in the ideal I(P) if and only if the bad player has a winning strategy in the game with the payoff set f^{−1}A. So the universally Baire part of the ideal I(P) is equal to its coanalytic part. Still another example of such a trivialization is provided by

Fact 2.17. [29] (ZFC+LC) For every ideal σ-generated by closed sets, every universally Baire positive set has a Borel positive subset.

It is also clear that for posets such as the Mathias forcing the hierarchy does not trivialize, since the Mathias forcing is a two step iteration whose first component is σ-closed. Thus the ideal associated with Mathias forcing cannot be directly analyzed by an integer game. It should be possible to isolate posets of this type, leading to

Question 2.18. Discern the parts of the creature machinery of [20] that can be analyzed by integer games.

3 Single step forcing: examples

In this Section I will consider several traditional or less traditional proper definable real forcings P and I will compute a σ-ideal I such that the poset P is forcing-equivalent to P_{I}. The obtained ideals will have two important properties:

First, I will be homogeneous meaning that for every I-positive Borel set B there is a function f : ℝ → B such that the preimages of I-small sets are I-small. The homogeneity means that for every Borel I-positive set B ⊂ ℝ it is the case that cov(I) = cov(I ↾ B) and non(I) = non(I ↾ B), where I ↾ B = {A ⊂ ℝ : A ∩ B ∈ I}. This is clear since the function f maps I-positive sets to I ↾ B-positive sets and pulls back I ↾ B-covering families to I-covering families. If the homogeneity is witnessed by Borel injections–which is the case for all ideals investigated in this section–then the factor forcing P_{I} must be homogeneous.

Second, except for the E_{0} and E_{1} forcings and the Mathias forcing the universally Baire part of the ideal I will be equal to its projective part. This will be proved using an integer game argument showing that under ZF+AD, every I-positive set has a Borel positive subset. It will mean that the forcing adds an essentially minimal real, and it will be also used in the analysis of the countable support iteration of the forcing using transfinite games in the next section.

3.1 Sacks forcing

The Sacks forcing is the partial order of all nonempty perfect trees in 2^{<ω} ordered by inclusion. The following dichotomy is relevant:

Fact 3.1. [7], 43.5. Every Borel set is either countable or it contains a perfect subset. In the context of AD this extends to all sets of reals.

Thus the Sacks forcing is isomorphic to a dense subset of the algebra B(2^{ω}) minus the countable sets ordered by inclusion, namely to the set of all perfect sets. Lemma 2.3 now shows that the Sacks forcing is proper–a proof quite dissimilar to the traditional argument. It is also clear that the ideal of countable sets is homogeneous. If B is an uncountable Borel set then it has a perfect subset C and there is a canonical continuous bijection f : ℝ → C. And for any bijection the preimages of countable sets are countable.

3.2 Miller forcing

The Miller forcing is the set of all superperfect trees ordered by inclusion. A superperfect tree is a nonempty subtree of ω^{<ω} such that every node in it has some prolongation with infinitely many immediate successors in the tree. The following dichotomy shows that Miller forcing is isomorphic to a dense subset of the algebra B(ω^{ω}) minus the σ-bounded sets.

Fact 3.2. [10] Every Borel subset of ω^{ω} is either σ-bounded or else it contains all branches of some superperfect tree. Under AD this extends to all subsets of ω^{ω}.

Lemma 2.3 now applies to show that Miller forcing is proper. It is quite easy to see that the ideal is homogeneous. If B ⊂ ω^{ω} is an unbounded set, then it contains all branches of some superperfect tree T. By further pruning of the tree T we can assume that every branching node of the tree T is actually infinitely branching. The reader can verify that the natural continuous bijection f : ω^{ω} → [T] ⊂ B has the property that preimages of bounded sets are bounded.

3.3 Laver forcing

The Laver forcing is the partial order of all Laver trees ordered by inclusion. A Laver tree T ⊂ ω^{<ω} is a nonempty tree with a trunk t ∈ T such that every longer sequence s ∈ T has infinitely many immediate successors. The following dichotomy shows that Laver forcing is a dense subset of positive Borel subsets of ω^{ω} with respect to the ideal I on ω^{ω} generated by the sets A_{g} = {f ∈ ω^{ω} : for infinitely many numbers n, f(n) ∈ g(f ↾ n)}, as g runs through all functions from ω^{<ω} to ω.

Lemma 3.3. [39] Every Borel set either belongs to the ideal I or it contains all branches of some Laver tree. In the context of AD this extends to all sets of reals.

Proof. Suppose A ⊂ ω^{ω} is a set and define an infinite game by letting players Adam and Eve play sequences t_{n} ∈ ω^{<ω} and bits b_{n} ∈ 2 respectively, observing the following rules: b_{0} = 1 and whenever Eve accepts a sequence t_{n}–that is, plays b_{n} = 1–then Adam submits one-step extensions t_{n+1}, t_{n+2}, … of t_{n} until Eve accepts one of them. The last number on the sequences t_{n+1}, t_{n+2}, … must increase. Adam wins if either Eve accepted only finitely many times or else ⋃{t_{n} : b_{n} = 1} ∈ A. The following two claims will complete the proof of the lemma by [16]:

Claim 3.4. Adam has a winning strategy if and only if the set A contains all branches of some Laver tree.

Proof. For the right to left direction fix a Laver tree T with [T] ⊂ A. Let Adam set t_{0} =trunk of T, and if t_{n} ∈ T has been played and accepted by Eve then let Adam submit immediate successors of the node t_{n} in the tree T in the increasing order until Eve accepts one of them. This is obviously a winning strategy for Adam.

For the left to right direction let σ be a winning strategy for Adam and let T ⊂ ω^{<ω} be the tree of all sequences that can possibly arise in a run of the game G_{A} in which Adam follows the strategy σ. Note that for each node t ∈ T there is a unique shortest run τ(t) such that it respects σ and t occurs in it, and if t ⊂ s are both in the tree T then τ(t) ⊂ τ(s). It follows that every branch f ∈ [T] is a result of the run ⋃{τ(t) : t ⊂ f} and therefore must belong to the set A. It is also clear from the definition of the game G_{A} that T is a Laver tree with trunk σ(0).

Claim 3.5. Eve has a winning strategy if and only if A ⊂ A_{g} for some function g : ω^{<ω} → ω.

Proof. For the right to left direction fix a function g such that A ⊂ A_{g}. Let Eve accept a sequence t_{n}, a one-step extension of some previously accepted sequence t_{m} as soon as the last number on t_{n} exceeds g(t_{m}). The result of such a play must fall outside of the set A_{g} and therefore this is a winning strategy for Eve.

For the left to right direction let σ be a winning strategy for Eve. For every sequence s ∈ ω^{<ω} let T_{s} be the tree of all sequences that can be accepted by Eve in some run of the game where he follows the strategy σ and Adam plays t_{0} = s. It follows that for all sequences s ⊂ t, if t ∈ T_{s} then all but finitely many one-step extensions of t must belong to the tree T_{s}–otherwise Adam could win by first getting to t and then submitting all the one-step extensions of t which do not belong to the tree T_{s}. Also, [T_{s}] ∩ A = 0 for all s ∈ ω^{<ω}. To see this, fix a branch f ∈ [T_{s}] and define S to be the tree of all partial runs of the game G_{A} in which Adam set t_{0} = s, Eve followed the strategy σ and the last move of Adam was accepted and it is an initial segment of the branch f. The tree S is ordered by extension. It follows from the “increasing” rule of the game G_{A} that the tree S is finitely branching–each run τ ∈ S has at most 2^{f(n)} immediate successors where n is the length of the last move of τ. Also, the tree S has height ω, so it must be illfounded. Any infinite branch of the tree S yields a run of the game G_{A} following the winning strategy σ whose result was the function f. Thus f ∉ A.

Now define a function g : ω^{<ω} → ω by setting g(t) = an integer such that for every s ⊂ t, if t ∈ T_{s} then g(t) is larger than all of the finitely many numbers i such that t^{⌢}i ∉ T_{s}. I claim that A ⊂ A_{g}. If this were not true then there would be a function f ∈ A such that for some n ∈ ω, for all larger numbers m necessarily g(f ↾ m) ∈ f(m). But then f ∈ [T_{f↾n}] by the definition of the function g, so by the previous paragraph f ∉ A. A contradiction!

The ideal I is clearly homogeneous. Whenever B ⊂ ω^{ω} is a Borel positive set, just choose a Laver tree T ⊂ ω^{<ω} such that [T] ⊂ B and consider the natural continuous bijection f : ω^{ω} → [T]. The preimages of I-small sets are I-small.

Again, there is a determinacy proof of the properness of Laver forcing. This time it is more complicated. It should be noted that the proof is really a generalization of the argument for Lemma 2.3 and that it gives a bit more information than needed. It is not clear whether the properness of every proper definable factor algebra can be proved along these lines.

Lemma 3.6. Laver forcing is proper.

Proof. I will show that given an arbitrary forcing P, a P -name ṙ for a real that is forced to fall out of all ground model coded I-small sets, a countable elementary submodel M of a large enough structure and a condition p ∈ M, the set {r ∈ ω^{ω} : for some M-generic filter G ⊂ P ∩ M with p ∈ G, r = ṙ/G} is I-positive. This fact applied to the special case of P_{I} gives the properness of this poset by 2.2 (2).

So let P and ṙ be as assumed in the previous paragraph. Define an infinite game between Adam and Eve in the following way. Adam starts, plays a condition p_{0} ∈ P and then on a fixed schedule produces one by one a sequence O_{n} : n ∈ ω of open dense subsets of the poset P and bit by bit a function g : ω^{<ω} → ω. Eve produces one by one a descending chain p_{n} : n ∈ ω of conditions in the poset P starting with p_{0}. Eve is allowed to tread water, that is, to wait for an arbitrary finite number of rounds before making her next move. Eve wins if the filter G generated by the chain she has constructed meets all the dense sets O_{n} : n ∈ ω and all the dense sets necessary for the evaluation of the name ṙ, and ṙ/G ∉ A_{g}. The key claim:

Claim 3.7. Eve has a winning strategy.

Given this claim, look at an arbitrary countable elementary submodel M of a large enough structure and a condition p ∈ M; the model will contain one of Eve’s winning strategies in the game, say σ. To show that the set X = {r ∈ ω^{ω} : for some M-generic filter G ⊂ P ∩ M with p ∈ G, r = ṙ/G} is I-positive, it is just necessary to verify that it is not a subset of any of the sets A_{g} for a function g : ω^{<ω} → ω. Simulate a run of the game in which Adam plays p = p_{0} and then enumerates the open dense subsets of the poset P in the model M and builds the function g, and Eve follows her winning strategy σ. All the moves in this game are in the model M and therefore the filter G ⊂ P Eve constructed is M-generic. Since Eve has won this run of the game, the real ṙ/G ∈ X falls out of the set A_{g} as desired.

The key observation for the proof of the claim is that the payoff set of the game is Borel on the (rather large) space of all the possible plays of the game and therefore the game is determined [17], [12] Theorem 20.5. Thus it is enough to derive a contradiction from the assumption that Adam has a winning strategy.

The following simple observation will be instrumental. For every condition p ∈ P there is a finite sequence s of integers which suits p. That is to say, for every function g : ω^{<ω} → ω there is a condition q ≤ p forcing that š ⊂ ṙ and for every j ≥ |s|, ǧ(ṙ ↾ j) ∈ ṙ(j). If this failed for some condition p, then for each sequence s there would be a counterexample g_{s}. Consider the diagonalization h of these functions, that is, h : ω^{<ω} → ω defined by h(t) = max{g_{s}(t) : s ⊆ t}. There must be a condition q ≤ p and a sequence s such that q forces š ⊂ ṙ and for all j ≥ |s|, ȟ(ṙ ↾ j) ∈ ṙ(j). This contradicts the choice of the function g_{s}.

Now consider a putative Adam’s strategy σ. Eve will construct a counterplay p_{0} ≥ p_{1} ≥ … together with sequences s_{0} ⊂ s_{1} ⊂ … so that the following are inductively satisfied:

• the sequence s_{i} suits the condition p_{i} • p_{i+1} ∈ O_{i} and p_{i+1} ⊩ š_{i} ⊂ ṙ

• at the stage where p_{i+1} has been played the strategy σ has already committed at the values of g(t) for all proper initial segments t of s_{i+1} and moreover for all numbers j with |s_{i}| ≤ j < |s_{i+1}|, g(s_{i+1} ↾ j) ∈ s_{i+1}(j).

In the end, the filter G ⊂ P generated by the descending chain of p_{i}’s meets all the necessary open dense sets and ṙ/G ∉ A_{g} because for all numbers j with |s_{0}| ≤ j necessarily g(ṙ/G) ↾ j) ∈ ṙ/G(j) by the last item above. Thus Eve has won against the strategy σ, contradicting the assumptions.

Now I just have to describe how Eve succeeds in maintaining the inductive hypothesis. First of all, she lets s_{0} be an arbitrary sequence such that s_{0} suits p_{0}. Once p_{i}, s_{i} have been constructed, Eve looks at the function g the strategy σ would produce if she just treaded water from this point on. Since s_{i} suits p_{i}, there is a condition p_{i+1} ∈ O_{n} such that p_{i+1} ⊩ š_{i} ⊂ ṙ and for all numbers j ≥ |s_{i}|, ǧ(ṙ ↾ j) ∈ ṙ(j). Also, Eve chooses a sequence s_{i+1} that suits p_{i+1}. Note that necessarily s_{i} ⊂ s_{i+1}. This determines the next Eve’s move, and she will play it only after the strategy σ has committed on the values of g(s_{i+1}) ↾ j for all numbers j ∈ |s_{i+1}|. By the choice of the condition p_{i+1} the third item above must be satisfied.

3.4 Cohen forcing

The Cohen forcing is the poset of finite sequences of integers ordered by reverse extension. The following dichotomy shows that this poset is forcing-equivalent to the algebra of Borel subsets of ω^{ω} minus the meager sets.

Fact 3.8. [7] Theorem 102(b). Every Borel set is either meager or there is a basic open set in which it is comeager. In the presence of AD this extends to all sets of reals.

In contradistinction to the previous cases, it is not the case that for every non-meager Borel set B there is an open set C such that C ⊂ B; the only thing we get is that there is an open set C such that C \ B is meager, so the equivalence class of the set C is below that of B in the separative quotient of the algebra of Borel subsets of ω^{ω} minus the meager sets.

To argue for the homogeneity of the meager ideal, let B be a Borel nonmeager set and choose a basic open set in which B is comeager; say that this is the open set of all reals beginning with a sequence s ∈ ω^{<ω}. Let f : ℝ → B be the map define by f(r) = s^{⌢}r if this belongs to the set B and f(r) =an arbitrary fixed element of the set B otherwise. It is easy to see that the preimages of meager sets are meager.

3.5 Solovay forcing

The Solovay forcing is defined as the algebra of Borel sets of reals minus the measure zero sets.

Fact 3.9. Under AD, every nonnull set of reals has a Borel nonnull subset.

Of course this follows from the standard covering game argument in [7] Theorem 102. The reader should note an alternative argument to show that every universally Baire nonnull set has a Borel nonnull subset: Just use the amoeba forcing to increase the additivity of the null ideal and consult Lemma 2.12 (2) and Lemma 2.13 (2).

The proof of the homogeneity of the measure zero ideal is left to the reader.

3.6 The E_{0} forcing

There seems to be as yet unexplored relationship between the forcing theory and the Borel equivalence relations. The partial order presented in this subsection is a very modest, but still illustrative beginning. First recall some basic notions of the theory of Borel equivalence relations:

Definition 3.10. An equivalence relation E on the reals is called smooth if there is a Borel function f : ℝ → ℝ such that for all reals x, y xEy if and only if f(x) = f(y). A set A ⊂ ℝ is called E-smooth if there is a Borel invariant set C ⊃ A and a Borel function f : C → ℝ such that for reals x, y ∈ C, xEy if and only if f(x) = f(y).

Definition 3.11. The symbol E_{0} denotes the Vitali equivalence on the reals: xE_{0}y if the difference x−y is a rational number. The E_{0}-smooth sets of reals will be called E_{0}-small, and the Borel non-E_{0}-smooth sets will be called E_{0}-positive.

It is a basic result of the theory of Borel equivalence relations that E_{0} is the simplest nonsmooth equivalence relation:

Fact 3.12. [15] [32] Let E be a Borel equivalence relation on the reals and A ⊂ ℝ be an analytic set. Exactly one of the following holds:

• A is E-smooth

• there is a one-to-one continuous function g : ℝ → A such that for all reals x, y xE_{0}y if and only if g(x)Eg(y).

Now consider the collection I of all E_{0}-small sets. It is not difficult to see that I is a σ-ideal. It follows from the Lusin-Novikov Theorem [12], 18.10, that the ideal I is σ-generated by partial Borel E_{0}-transversals, that is, Borel sets which meet every E_{0}-class in at most one point.

Lemma 3.13. The partial order P_{I} is proper.

Proof. The argument begins with a small observation.

Claim 3.14. Let B ∈ P_{I} be a Borel E_{0}-positive set. There exists a nonzero rational number q ∈ ℚ such that the set {r ∈ B : r + q ∈ B} is I-positive.

If this failed for a positive Borel set B then all of the sets A_{q} = {r ∈ B : r + q ∈ B} are E_{0}-small for nonzero rational numbers q, and the set B \ ⋃_{q} A_{q} is a partial Borel selector, therefore E_{0}-small. Thus the E_{0}-positive set B would be a union of countably many E_{0}-small sets, contradiction.

Let M be a countable elementary submodel of a large enough structure. It must be argued that for every set B ∈ P_{I} ∩ M the set {r ∈ B : r is M-generic for P_{I}} is E_{0}-positive. The key claim:

Claim 3.15. Let B ∈ P_{I} ∩ M be an E_{0}-positive set and let {O_{n} : n ∈ ω} be a countable collection of open dense subsets of the poset P_{I} ∩ M. There are two distinct M-generic reals r_{0}, r_{1} ∈ B such that the respective filters G_{0} = {C ∈ P_{I} ∩ M : r_{0} ∈ C} and G_{1} = {C ∈ P_{I} ∩ M : r_{1} ∈ C} meet all these open dense sets, and such that the difference r_{0} − r_{1} is a rational number.

Proof. Without loss of generality assume that the list {O_{n} : n ∈ ω} contains all the sets O ∩ M where O ∈ M is an open dense subset of the forcing P_{I}. First use the previous claim to find a nonzero rational number q such that B_{0} = {r ∈ B : r + q ∈ B} is nonsmooth. Since the smooth ideal is translation invariant, it is clear that both sets B_{0} and C_{0} = B_{0}+q are nonsmooth conditions in the forcing P_{I} ∩ M stronger than B. Now by induction on n ∈ ω construct sets B_{n}, C_{n} so that

1. B_{0} ⊃ B_{1} ⊃ B_{2} ⊃ … and C_{0} ⊃ C_{1} ⊃ C_{2} … are chains in the ordering P_{I} ∩ M

2. B_{n+1}, C_{n+1} ∈ O_{n}

3. C_{n} = B_{n} + q.

Once this is done, look at the intersections ⋂_{n} B_{n} and ⋂_{n} C_{n}. Since the chains are M-generic, both of them contain a single real, r_{0} and r_{1} respectively. But by the third item above, r_{0} + q ∈ ⋂_{n} C_{n} and therefore r_{1} = r_{0} + q. The claim follows.

To perform the induction, suppose that B_{n} has been obtained and C_{n} = B_{n} + q. Let B̄_{n} ⊂ B_{n} be a condition in the set O_{n}. The set C̄_{n} = B̄_{n} + q is a condition stronger than C_{n}. Let C_{n+1} be a condition in the set O_{n} stronger than C̄_{n} and let B_{n+1} = C_{n+1} − q.

The claim really shows that for every partial Borel E_{0}-transversal A ⊂ ℝ the set of all filters g ⊂ P_{I} ∩ M for which ṙ_{gen}/g ∈ A, is meager. Since if it were not, it would be comeager under some condition B ∈ P_{I} ∩ M and then the claim could be used to produce two distinct reals in the set A with rational difference, contradicting the assumption on the set A.

Now the lemma quickly follows. Suppose that B ∈ P_{I} ∩ M is an E_{0}-positive Borel set. I must show that the set C = {r ∈ B : r is M-generic} is E_{0}-positive. Well, suppose that {A_{n} : n ∈ ω} are partial Borel E_{0}-transversals; it will be enough to produce a real in C \ ⋃_{n} A_{n}. But by the previous paragraph there are even comeagerly many filters g ⊂ P_{I} containing the condition B for which the real ṙ_{gen} falls out of all sets {A_{n} : n ∈ ω}, so one of them certainly produces the required M-generic real.

Lemma 3.16. The ideal I is homogeneous.

Proof. This is implied by the Glimm-Effros dichotomy 3.12. Suppose that B ⊂ ℝ is a E_{0}-positive Borel set. Then there is a one-to-one continuous function g : ℝ → B such that for all reals x, y xE_{0}y if and only if g(x)E_{0}g(y). It is not difficult to see that if A ⊂ B is an E_{0}-small set (as witnessed by a set C and a function f) then its preimage g^{−1}A is E_{0}-small too (as witnessed by the set f^{−1}C and the function f ∘ g).

It is not quite clear how the forcing P_{I} relates to the traditional forcing notions. It is not hard to see that it is nowhere c.c.c. The argument in the proof of Theorem 5.28 can be used to show that in the iterated P_{I}-extension 𝔞 = 𝔡 = 𝔱 = ℵ_{1}, in particular P_{I} is bounding. However, there is a distinction between P_{I} and the previous examples:

Fact 3.17. [2], Theorem 2. (ZFC+LC) The universally Baire part of the ideal I(P_{I}) is properly larger than its Borel or projective part.

3.7 The E_{1} forcing

It is not always possible to generalize the approach of the previous section to other situations where dichotomy theorems are known.

Definition 3.18. [11]

A Borel equivalence is hypersmooth if it is an increasing union of countably many Borel smooth equivalence relations.

Definition 3.19. [11] The symbol E_{1} stands for the tail equivalence on ℝ^{ω}. Thus sequences r⃗, s⃗ are E_{1}-equivalent if there is an integer n such that for all larger m, r⃗(m) = s⃗(m).

It is not difficult to see that both E_{0} and E_{1} are hypersmooth equivalence relations, E_{0} is reducible to E_{1} and E_{1} is not reducible to E_{0}. In fact, every hypersmooth equivalence relation can be reduced to E_{1} and E_{1} cannot be reduced to any equivalence relation with countable classes. The following is a basic result of the Borel equivalence relation theory.

Fact 3.20. [11] Let E be a Borel hypersmooth equivalence relation. Then

• either E is reducible to E_{0}

• or E_{1} is reducible to E.

So the Borel subsets of ℝ^{ω} divide into two classes: the E_{1}-positive sets B, that is those for which E_{1} is reducible to E_{1} ↾ B, and E_{1}-small sets B for which E_{1} ↾ B is reducible to E_{0}. Let I be the collection of all subsets of ℝ^{ω} with a Borel E_{1}-small superset. It is not difficult to see that I is a σ-ideal, and a question may be raised, is the forcing P_{I} proper? In this case, I will give a negative answer. First, a preliminary lemma of independent interest.

Lemma 3.21. Let L(ℝ)(Γ) be an inner model of Real Determinacy and Dependent Choice all of whose sets of reals are universally Baire. In this model, the collection J of all subsets of ℝ^{ω} with no E_{1}-positive Borel subset is an ordinal covering σ-ideal on ℝ^{ω}.

Proof. In the model L(ℝ)(Γ) let Ī be the smallest ordinal-covering σ-ideal containing I as a subset. I will show that whenever B is an E_{1}-large Borel set, then B ∉ Ī. It follows that J = Ī. Clearly Ī ⊂ J because every set not in J has an E_{1}-large Borel subset and therefore does not belong to Ī. On the other hand, J ⊂ Ī, since every Ī-positive set contains a Ī-positive Borel subset, such a set must be E_{1}-large and therefore the original set was not in J. The lemma follows.

Consider the reverse iteration P of Sacks forcing along the inverted ω ordering, adding a generic sequence r⃗_{gen} ∈ ℝ^{ω}, as described in Subsection 5.4. I will show that the P -generic sequence belongs to no ground model coded Borel E_{1}-small set. For suppose that p ∈ P and B are a condition and an E_{1}-small Borel set such that p ⊩ r⃗_{gen} ∈ Ḃ, and suppose that g : B → ℝ is a Borel function reducing E_{1} ↾ B to E_{0}. Let M be a countable elementary submodel of a large enough structure containing the condition p and the set B. The work in Subsection 5.4 provides for the following:

• There is a condition q ≤ p consisting solely of M-generic sequences in ℝ^{ω}. Clearly, it must be the case that q ⊂ B.

• There is a projection-keeping homeomorphism f : ℝ^{ω} → q. Such a homeomorphism reduces E_{1} to E_{1} ↾ q.

• The function g ∘ f reduces E_{1} to E_{0}, contradiction.

Now work in the model L(ℝ)(Γ). Let B be an E_{1}-positive set, and let f : ℝ^{ω} → B be a Borel function reducing E_{1} to E_{1} ↾ B. I must show that B ∉ Ī. Let K = {A ⊂ ℝ^{ω} : P ⊩ ¬ḟ(r⃗_{gen}) ∈ Ȧ}. By Lemma 2.12 (1), this is an ordinal-covering σ-ideal. It is enough to show that I ⊂ K, because then B ∉ K ⊃ Ī by the minimality of the ideal Ī. But I ⊂ K is clear: if C ⊂ B is an E_{1}-small Borel set then even f^{−1}C is E_{1}-small, and by the previous paragraph the poset P forces the generic sequence to fall out of it.

Now it is possible to give a concise proof of the improperness of the poset P_{I}.

Lemma 3.22. The forcing P_{I} is improper.

Proof. The forcing P_{I} adds a generic sequence r⃗_{gen} of reals. The key to the proof is the following claim:

Claim 3.23. P_{I} forces that for every integer n there is a ground model Borel function f such that r⃗_{gen} = f(r⃗_{gen} \ n).

Here for an ω-sequence r⃗ of reals and an integer n, r⃗ \ n is the sequence s⃗ given by s⃗(i) = r⃗(n + i) for every integer i ∈ ω.

To prove the claim, let B ∈ P_{I} be an arbitrary condition and n ∈ ω be an integer. The set C = {r⃗ \ n : r⃗ ∈ B} must be J-positive: it is analytic, therefore in L(ℝ) a union of ℵ_{1} many Borel sets {C_{α} : α ∈ ω_{1}}, and if each of these sets were E_{1}-small then so would be the sets B_{α} : α ∈ ω_{1} defined by r⃗ ∈ B_{α} if r⃗ \ n ∈ C_{α}. Since the set B is a subset of ⋃_{α} B_{α}, this would contradict the closure of the ideal J under wellordered unions in L(ℝ) proved in the previous lemma. By Lemma 2.13 (3) there is a Borel E_{1}-large subset D ⊂ C and a Borel function f : D → B such that for every sequence s⃗ ∈ D, s⃗ = f(s⃗) \ n. Clearly, the function f is one-to-one and its range E ⊂ B is a Borel E_{1}-positive set forcing that r⃗_{gen} = ḟ(r⃗_{gen} \ n).

The lemma immediately follows. Let M be a countable elementary submodel of a large enough structure. It is enough to show that the set C of all M-generic sequences is E_{1}-small, and for this it is enough to show that the E_{1}-equivalence classes restricted to the set C are at most countable, because E_{1} cannot be reduced to any countable equivalence relation. But the claim shows that given an M-generic sequence r⃗, all of the M-generic sequences E_{1}-equivalent to r⃗ are included in the countable set {f(r⃗ \ n) : f ∈ M, n ∈ ω}.

In fact, the poset P_{I} collapses 𝔠 to ω.

3.8 The c_{min} forcing

It is obvious from the previous two subsections that dichotomy theorems frequently give rise to homogeneous ideals. The homogeneity is a quite important property of an ideal–see for instance its use in the statements and proofs of the duality and isolation theorems in the Applications section. I will now give another example of the connection between dichotomies and homogeneity.

Let c_{min} : [2^{ω}]^{2} → 2 be the continuous coloring defined by c_{min}(x, y) = Δ(x, y) mod 2 where Δ(x, y) is the least number n ∈ ω such that x(n) ≠ y(n). Let K be the ideal σ-generated by the homogeneous sets for this coloring. It is not difficult to see that this is indeed a proper σ-ideal. Moreover, c_{min} is the minimal continuous coloring with this property:

Lemma 3.24. [30] Suppose that X is a Polish space, d : [X]^{2} → 2 is a continuous coloring and B ⊂ X is an analytic set. Then

• either B is a union of countably many homogeneous sets

• or there is a topological embedding f : 2^{ω} → B such that for all x, y ∈ 2^{ω}, c_{min}(x, y) = d(f(x), f(y)).

The conclusion:

Lemma 3.25. K is a homogeneous ideal σ-generated by closed sets. Thus the forcing P_{K} is proper and homogeneous.

Proof. For the homogeneity of the ideal K, look at an arbitrary positive Borel set B ⊂ 2^{ω} and use the previous lemma to find a Borel injection f : 2^{ω} → B respecting the coloring c_{min}. Clearly, the preimages of K-small sets are K-small. Finally, since the coloring c_{min} is continuous, the closure of a homogeneous set is homogeneous, and so the ideal K is σ-generated by closed sets.

Similar theorems hold for the colorings c_{min}^{n} : [2^{ω}]^{2} → n defined by c_{min}^{n} = Δ(x, y) mod n and the ideal K_{n} σ-generated by the sets avoiding at least one color.

3.9 Mathias forcing

The Mathias forcing is the partial order of all pairs ⟨s, a⟩ where s is a finite set of integers and A is an infinite set of integers with max(s) ∈ min(a) ordered by ⟨s, a⟩ ≥ ⟨t, b⟩ if s ⊂ t, b ⊂ a and t \ s ⊂ a. The following observation shows that this poset is forcing-equivalent with P_{I} where I is the ideal of set nowhere dense in the algebra P(ω) minus the ideal of finite sets.

Lemma 3.26. If A ⊂ P is a universally Baire set then either A is nowhere dense in the algebra P(ω) minus the ideal of finite sets, or there is a condition in the Mathias forcing which forces the generic real into Ȧ.

Proof. The argument depends on the well-known geometric property of Mathias forcing:

Fact 3.27. [27] If M is a transitive model of set theory and a ⊂ ω is an Mgeneric set for the Mathias forcing then every infinite set b contained in a up to finitely many elements is M-generic for the Mathias forcing as well.

Now let A be a universally Baire set, A = p[T] for some universally Baire tree T. Suppose first that some condition p = ⟨s, c⟩ forces the generic set into p[Ť]. Choose a countable elementary submodel M of a large enough structure containing all the important information. Choose an M-generic set a ⊂ ω for the Mathias forcing, meeting the condition p, meaning that a ⊂ s ∪ c and s ⊂ a. By the forcing theorem, M[a] ⊧ a ∈ p[T], and so a ∈ p[T]. Now also every infinite subset b ⊂ a with s ⊂ b will be M-generic meeting the condition p by the Fact 3.27, and therefore b ∈ p[T]. Thus A = p[T] is dense below the set a in the algebra P(ω) minus the ideal finite sets.

For the other direction, suppose that A is dense below some infinite set c ⊂ ω. Let M be a countable elementary submodel containing all the necessary information, in particular c, A ∈ M. Let a ⊂ c be an M-generic Mathias set and let b ∈ A be an infinite set such that b ⊂ a up to finitely many elements. Then a is M-generic by Fact 3.27 again, by the universally Baire absoluteness M[a] ⊧ a ∈ A and so by the forcing theorem there must be a condition in the Mathias forcing that forces the generic set to be an element of Ȧ.

I do not see any determinacy proof of the properness of Mathias forcing. The reason may be that the universally Baire part of the associated ideal is not projectively generated-Lemma 2.15.

Question 3.28. Is there a partial order P adding an infinite subset ȧ of ω such that P forces ȧ to fall out of all ground model coded universally Baire sets that are nowhere dense in the algebra P(ω) minus the finite sets, yet for every countable elementary submodel M of large enough structure the set {b ⊂ ω : b is M-generic for P } is nowhere dense?

4 The countable support iteration

The results of Section 2 raise many questions. Since there is such a tight relationship between real proper forcings and ideals, there should be a correspondence between the properties of ideals and properties of forcing notions, as well as between operations on ideals and operations on forcing notions. In particular, the various notions of iteration and product of posets should be expressible using some operations on ideals, and the countable support iteration is the natural first subject of study.

Let P be a proper definable real forcing. In the first subsection, I will provide a geometric representation of the countable support iteration of the poset P, which is much better suited for the applications than the traditional one. The second subsection introduces the transfinite Fubini powers of σ-ideals. The last two subsections contain the critical dichotomies which show how the Fubini powers are related to the iteration.

4.1 The geometric view of the iteration

The basic geometric notion connected to the countable support iteration is the following.

Definition 4.1. Let α be a countable ordinal and I a σ-ideal on the reals. A Borel I-perfect set is a set B ⊂ ℝ^{α} such that

• for every ordinal β ≤ α the set B ↾ β = {r⃗ ↾ β : r⃗ ∈ B} is Borel

• for every ordinal β ∈ α and every sequence s⃗ ∈ B ↾ β the set {r ∈ ℝ : s^{⌢}r ∈ B ↾ β + 1} is I-positive

• for every increasing sequence β_{n} : n ∈ ω of ordinals below α and every inclusion increasing sequence s⃗_{n} : n ∈ ω of sequences s⃗_{n} ∈ B ↾ β_{n}, it is the case that ⋃_{n} s⃗_{n} ∈ B ↾ ⋃_{n} β_{n}.

There are two degrees of freedom in the above definition. For a Wadge pointclass Γ I will speak of the Γ I-perfect sets in the obvious sense, and I will also use a similarly defined notion of a Borel I-perfect subsets of ℝ^{X} for a countable set X of ordinals.

It seems that a reformulation of the countable support iteration of a proper definable real forcing using Borel I-perfect sets has been long known, but neglected. It figures prominently in Kanovei’s [9]. Let P be a proper definable real forcing and κ an ordinal. Define the poset P_{κ} to consist of all Borel I(P)-perfect subsets B of ℝ^{X} for all countable subsets X of κ. The set X will be called the domain of B, dom(B). The ordering is defined by C ≤ B if dom(B) ⊂ dom(C) and for every sequence r⃗ ∈ C, r⃗ ↾ dom(B) ∈ B.

Lemma 4.2. (ZFC+LC) Let P be a provably proper definable real forcing and κ an ordinal. The poset P_{κ} is forcing isomorphic to the countable support of length κ of P.

Proof. Fix the poset P and identify it with P_{I(P)}. The lemma uses a technical claim saying that P must have natural master conditions for many countable models, not necessarily elementary. Ther reader should compare the claim with Shelah’s snep definitions [24].

Claim 4.3. (ZFC+LC) Let M be a countable elementary submodel of a large enough structure, Q ∈ M a partial order, g ⊂ M ∩ Q an M-generic filter, and B ∈ P ∩ M[g] a condition. The set C = {r ∈ B : r is M[g]-generic for the poset P } is I(P)-positive.

Proof. Fix the model M and the filter g and for the sake of Simplicity identify the model M[g] with its transitive collapse. Let δ be a Woodin cardinal of M[g], and in V find an M[g]-nonstationary tower M[g]-generic filter h and the corresponding embedding j : M[g] → N. Now P ∩ M[g][h] = P^{M[g][h]} since the poset P is universally Baire and M[g][h] is a generic extension of an elementary submodel of a large structure in V. Also P^{N} = P^{M[g][h]} since the models N and M[g][h] have the same reals. Choose an M[g]-inaccessible cardinal λ ∈ δ and look into the model N. Since N ⊧ j(P^{M[g]}) = P is proper, N ⊧ C = {r ∈ B : r is P -generic for the model j′′(M[g] ∩ V_{λ})} is I(P)-positive. Since the model M[g][h] has the same reals as N, also M[g][h] ⊧ C is I(P)- positive. Since the ideal I(P) is obtained by a real game quantifier, its Borel part is universally Baire by Fact C.8 and C.3 and so the set C is I(P)-positive even in V as desired.

Now look at the partial ordering P_{κ} and the countable support iteration R_{κ} of the poset P, and denote by r⃗_{gen} the canonical name for the R_{κ}-generic κsequence of reals. Define a map π : P_{κ} → R_{κ} by setting π(B) to be the canonical condition p ∈ R_{κ} with dom(p) = dom(B) and for every ordinal α ∈ dom(p) p ↾ α ⊩ p(α) = {r ∈ ℝ : (r⃗_{gen} ↾ dom(p) ∩ α)^{⌢}r ∈ Ḃ ↾ α + 1}. It is not difficult to see that π is an order-preserving map and π(B) ⊩ r⃗_{gen} ↾ dom(B) ∈ Ḃ. To prove the lemma, it is enough to show that π preserves incompatibility and the range of π is dense in R_{κ}. This will follow from the following claim.

Claim 4.4. Let M be a countable elementary submodel of a large enough structure, and p ∈ R_{κ} ∩ M. There is a Borel I(P)-perfect set B with domain M ∩ κ consisting only of M-generic sequences which meet the condition p. For every such a set B, π(B) ≤ p.

Proof. The argument for the claim is just a repetition of the proof of the preservation of properness under countable support iteration [26], Section III.3.3. Fix the model M and by induction on α ∈ M ∩ κ + 1 prove the following statement. For every ordinal β ∈ α ∩ M, every condition p ∈ P_{κ} ∩ M and every Borel I(P)- perfect set A with dom(A) = M ∩ β consisting only of M-generic sequences for the poset P_{β} which meet the condition p ↾ β, there is a Borel I-perfect set B with dom(B) = M ∩ α consisting only of M-generic sequences for the poset P_{α} which meet the condition p ↾ α, and such that A = B ↾ β. At the same time prove that every such set B satisfies π(B) ≤ p ↾ α.

The argument is essentially identical to the proof in [26] and its details are left to the reader. Let me just indicate the situation in which the previous claim is needed. It is in the successor stage of the induction. Suppose that α = β +1 ∈ M ∩κ are ordinals, p ∈ P_{κ} ∩M is a condition and A is a Borel I(P)- perfect set with dom(A) = M ∩β consisting only of M-generic sequences for the poset P_{β} which meet the condition p ↾ β. Then the set B = {r⃗ ∈ ℝ^{M∩α} : r⃗ is an M-generic sequence for the poset P_{α} meeting the condition p ↾ α and r⃗ ↾ β ∈ A} is a Borel I-perfect set and B ↾ β = A. The only nontrivial observation in the proof of this statement is that for every sequence s⃗ ∈ A the set {r ∈ p(β)/s⃗ : r is M[s⃗]-generic for the poset P^{M[s⃗]}} is I(P)-positive. But this is exactly the contents of the previous claim.

It is now clear that the range of the function π is dense in R_{κ}. To show that π preserves incompatibility, suppose that B, C ∈ P_{κ} are conditions and π(B), π(C) have a common lower bound p ∈ R_{κ}. Choose a countable elementary submodel M of a large structure containing all the information and use the claim to find a Borel I(P)-perfect set D consisting only of M-generic sequences meeting the condition p. Thus every sequence r⃗ ∈ D must meet both conditions π(B) and π(C) and by the Borel absoluteness between the model M[r⃗] and V it must be the case that r⃗ ↾ dom(B) ∈ B and r⃗ ↾ dom(C) ∈ C. This is to say, the set D is the desired common lower bound of the conditions B, C in the poset P_{κ}.

The proof actually gives several other pieces of information.

Corollary 4.5. Let α be a countable ordinal. If {A_{n} : n ∈ ω} are Borel subsets of ℝ^{α} without a Borel I(P)-perfect subset, then even their union does not contain such a subset.

Proof. This is true simply because a Borel subset of ℝ^{α} contains a Borel I(P)- perfect subset if and only if it is I(R_{α})-positive where R_{α} is the countable support iteration of the poset P of length α and the ideal is computed using the canonical generic sequence r⃗_{gen}. If a Borel set B ⊂ ℝ^{α} contains a Borel I(P)- perfect subset C then clearly π(C) ⊩ r⃗_{gen} ∈ Ċ ⊂ Ḃ and so B is I(P_{α})-positive by Lemma 2.10. On the other hand, if a Borel set B ⊂ ℝ^{α} is I(P_{α})-positive, then there must be a condition p ∈ R_{α} forcing r⃗_{gen} ∈ Ḃ. Find a countable elementary submodel M of a large enough structure containing α, p and B, and use Claim 4.4 to find a Borel I(P)-perfect set C consisting solely of M-generic sequences such that π(C) ≤ p. By the forcing theorem applied in the model M, for every sequence r⃗ ∈ C M[r⃗] ⊧ r⃗ ∈ B, by an absoluteness argument r⃗ ∈ B, and C ⊂ B as desired.

Corollary 4.6. Let α be a countable ordinal. Every universally Baire I(P)- perfect subset of ℝ^{α} has a Borel I(P)-perfect subset.

Proof. Fix the ordinal α and the universally Baire I(P)-perfect set B. By induction on β ∈ α construct a condition p ∈ R_{α} so that p ↾ β ⊩ r⃗_{gen} ↾ β ∈ Ḃ ↾ β for all β ∈ α + 1. The I(P)-perfectness of the set B implies that the induction hypothesis survives at limit stages. At the successor stages, use the universally Baire absoluteness to show that p ↾ β ⊩ {s ∈ ℝ : (r⃗_{gen})^{⌢}s ∈ Ḃ ↾ β̌ + 1} is a universally Baire I(P)-positive set and therefore has a Borel I(P)-positive subset. Then let p(β) be some R_{β}-name for such a Borel I(P)-positive subset, which is an element of the forcing P in the R_{β}-extension.

Now as in Claim 4.4 let M be a countable elementary submodel of a large enough structure containing the condition p, and let C be a Borel I-perfect subset of ℝ^{α} consisting solely of M-generic sequences meeting the condition p. The choice of the condition p and the forcing theorem applied in the model M imply that for each sequence r⃗ ∈ C, M[r⃗] ⊧ r⃗ ∈ B; by an absoluteness argument then, r⃗ ∈ B. Thus C ⊂ B is the desired Borel I(P)-perfect subset.

Let r⃗_{gen} be the P_{κ}-name for a κ-sequence of reals defined by r⃗_{gen}(α)(n) = m if and only if the set B_{α,n,m} = {⟨α, r⟩ : r ∈ ℝ, r(n) = m} ⊂ ℝ^{{α}} belongs to the generic filter.

Corollary 4.7. P_{κ} forces that for every set B ∈ P̌_{κ}, B is in the generic filter if and only if r⃗_{gen} ↾ dom(B) ∈ Ḃ.

Proof. First, it is necessary to prove that for every ordinal α ∈ κ and every integer m ∈ ω the sets {B_{α,n,m} : m ∈ ω} form a maximal antichain. An antichain they are, since they are pairwise disjoint. Now suppose C ∈ P_{κ} is an arbitrary condition. Strengthening the condition if necessary we may assume that α ∈ dom(C). Let C_{m} = {r⃗ ∈ C : r⃗(α)(n) = m}. By the previous Corollary, the set of all Borel subsets of ℝ^{dom(C)} without a Borel I(P)-perfect subset forms a σ-ideal basis, so one of the sets C_{m} must contain such a subset D. Then D is a lower bound of the set C and B_{α,n,m}, showing that the collection {B_{α,n,m} : m ∈ ω} is predense as desired.

For the left-to-right direction of the corollary argue that every condition B forces r⃗_{gen} ↾ dom(B) ∈ Ḃ. This will follow if I show that for all countable subsets X ⊂ Y of κ, all Borel sets B ⊂ ℝ^{X} and all Borel I(P)-perfect sets C ⊂ ℝ^{Y} such that C ↾ X ⊂ B, C ⊩ r⃗_{gen} ↾ X̌ ∈ Ḃ. This is proved by induction on the Borel complexity of the Borel set B ⊂ ℝ^{X}. It is clearly true for basic open and closed sets. If B = ⋂_{n} B_{n} and C is a Borel I(P)-perfect set such that C ↾ dom(B) ⊂ B then by the induction hypothesis C ⊩ r⃗_{gen} ↾ X ∈ ⋂_{n} Ḃ_{n} = Ḃ as desired. If B = ⋃_{n} B_{n}, C ∈ P_{κ} is such that C ↾ dom(B) ⊂ B and D ≤ C is an arbitrary condition, then one of the Borel sets {r⃗ ∈ D : r⃗ ↾ dom(B) ∈ B_{n}} : n ∈ ω must contain a Borel I(P)-perfect set D_{n}, and by the induction hypothesis D_{n} ⊩ r⃗_{gen} ↾ dom(B) ∈ Ḃ_{n}. Thus C ⊩ r⃗_{gen} ↾ dom(B) ∈ Ḃ as desired.

For the right-to-left direction first argue that if X ⊂ κ is a countable set and C is an I(P)-perfect subset with X ⊂ dom(C) then the set C ↾ X = {r⃗ ↾ X : r⃗ ∈ C} contains a Borel I(P)-perfect subset. This does not follow trivially from Corollary 4.6 because the set C ↾ X is not necessarily I(P)-perfect–the closure condition may fail. To overcome this difficulty, enumerate the ordinals in the set X in increasing order by γ_{β} : β ∈ α and by induction on the ordinal β ≤ α construct universally Baire I(P)-perfect sets D_{β} ⊂ ℝ^{X∩γ_{β}} and universally Baire functions f_{β} : D_{β} → C ↾ γ_{β} such that f_{β}(r⃗) ↾ X∩β = r⃗ for all sequences r⃗ ∈ D_{β}– so f(r⃗) is a witness to the fact that r⃗ ∈ C ↾ X ∩ γ_{β}. The sets D_{β} and functions f_{β} will cohere in the following sense: D_{δ} = D_{β} ↾ γ_{δ} and f_{δ}(r⃗ ↾ γ_{δ}) = f_{β}(r⃗) ↾ γ_{δ} for all sequences r⃗ ∈ D_{β}. In the end, D_{α} will be a universally Baire I(P)-perfect subset of the set C ↾ X and Corollary 4.6 will produce the desired Borel I(P)- perfect subset of it. The inductive construction proceeds as follows. At limit ordinals β take D_{β} = {r⃗ ∈ ℝ^{X∩γ_{β}} : ∀δ ∈ β r⃗ ↾ γ_{δ} ∈ D_{δ}} and for every sequence r⃗ ∈ D_{β} take f_{β}(r⃗) = ⋃_{δ∈β} f_{δ}(r⃗ ↾ γ_{δ}). At successor stages observe that for every sequence r⃗ ∈ D_{β} the set B(r⃗) = {s ∈ ℝ : ∃u⃗ ∈ C u⃗ ↾ γ_{β} = f_{β}(r⃗) ∧ u⃗(γ_{β+1}) = s} is Borel and I(P)-positive. Let D_{β+1} = {t⃗ ∈ ℝ^{X∩γ_{β+1}} : t⃗ ↾ γ_{β} ∈ D_{β} ∧ t⃗(γ_{β}) ∈ B(t⃗ ↾ γ_{β})} and use universally Baire uniformization C.3 to find a universally Baire function f_{β+1} with the required properties.

It is now clear that if X ⊂ κ is a countable set and B ⊂ ℝ^{X} is a Borel set without I(P)-perfect subset then P_{κ} ⊩ r⃗_{gen} ↾ X̌ ∉ Ḃ. To see this, let C ∈ P_{κ} be an arbitrary condition, and strengthening if necessary assume that X ⊂ dom(C). The set {r⃗ ∈ C : r⃗ ↾ X ∈ B} cannot contain a Borel I(P)-perfect subset, because if it did, the previous paragraph could be used to produce a Borel I(P)-perfect subset of the set B, contradicting the assumptions on B. Thus by Corollary 4.5, it must be the case that the set {r⃗ ∈ C : r⃗ ↾ X ∉ B} has a Borel I(P)-perfect subset D. By the Σ_{1}^{1} absoluteness and the second paragraph of this proof, D ≤ C is a condition forcing r⃗_{gen} ↾ X̌ ∉ Ḃ as desired.

Finally, for the right-to-left direction let B, C be arbitrary conditions in P_{κ} such that C ⊩ ṙ_{gen} ↾ dom(B) ∈ Ḃ. I need to find a common lower bound of the conditions B, C. Strengthening the condition C if necessary, assume that dom(B) ⊂ dom(C). The set D = {r⃗ ∈ C : r⃗ ↾ dom(B) ∈ B} must contain a Borel I(P)-perfect subset E–if it did not, it would be a Borel set without an I(P)-perfect subset such that C ⊩ ṙ_{gen} ↾ dom(C) ∈ Ḋ, contradicting the previous paragraph. Clearly, the set E is a common lower bound of the conditions B and C.

Of course the just defined name for P_{κ}-generic sequence of reals is moved by the embedding π into the canonical name for the R_{κ}-generic sequence, as Claim 4.4 shows.

Corollary 4.8. For every condition B ∈ P_{κ} and every P_{κ} name ẋ for a real there is a stronger condition C ≤ B and a Borel function f : C → ℝ such that C ⊩ ẋ = ḟ(r⃗_{gen} ↾ dom(Ċ)).

Proof. Fix the condition B and the name ẋ and choose a countable elementary submodel M of a large enough structure. Use Claim 4.4 to find a Borel I(P)- perfect set C with dom(C) = M ∩ κ consisting solely of M-generic sequences meeting the condition B. By the previous Corollary, C ≤ B. Let f : C → ℝ be the Borel function defined by f(r⃗) = ẋ/r⃗. An absoluteness argument shows that C ⊩ ẋ = ḟ(r⃗_{gen} ↾ dom(C)).

The rest of this section will be occupied by the precise calculation of the universally Baire part of the ideal I(P_{α}) for countable ordinals α. It turns out that this is exactly the collection of those universally Baire subsets of ℝ^{α} which do not have a Borel I(P)-perfect subset. However, the proofs will yield much more information.

4.2 Transfinite Fubini powers of ideals

The operation on ideals corresponding to the countable support iteration is the following.

Definition 4.9. For a countable ordinal α and a σ-ideal I on the reals, the α-th Fubini power of the ideal I, denoted by I^{α}, is the ideal on ℝ^{α} defined as the collection of all sets A ⊂ ℝ^{α} for which Adam has a winning strategy in the game G(A). The game, played between Adam and Eve, lasts α rounds. At round β ∈ α first Adam plays a set Y_{β} ∈ I and then Eve produces a number r_{β} ∈ ℝ \ Y_{β}. Eve wins if the sequence ⟨r_{β} : β ∈ α⟩ belongs to the set A.

It is quite easy to see that I^{α} must be a σ-ideal since Adam can cumulate his strategies. The definition gives simply the traditional Fubini powers of the ideal I at finite ordinals α; it is the infinite ordinals that make the definition interesting. The ideals I^{α} factorize in a natural way. If β ∈ α are countable ordinals then I^{α} is naturally isomorphic to the Fubini product of the ideal I^{β} and I^{α\β} in the sense that a set C ⊂ ℝ^{α} is I^{α}-positive if and only if the set {s⃗ ∈ ℝ^{β} : {t⃗ ∈ ℝ^{α−β} : s^{⌢}t⃗ ∈ A} ∉ I^{α−β}} is I^{β}-positive.

Given a proper definable real forcing P, the Fubini power (I(P))^{α} is the natural first attempt to compute the ideal I(P_{α}). It does not quite work, but at the same time it is quite successful: it will be shown in the following two subsections that the universally Baire parts of the two ideals do coincide. Here I will just make several observations on how the cardinal coefficients of the Fubini powers of the ideal I relate to those of I. The situation is quite clear in the cov and add case.

Lemma 4.10. Let I be a projectively generated σ-ideal and α ∈ ω_{1} a nonzero ordinal. Then cov(I) = cov(I^{α}) and add(I) = add(I^{α}).

Proof. The inequalities cov(I) ≤ cov(I^{α}) and add(I) ≤ add(I^{α}) are immediate and left to the reader. The opposite inequalities are handled below.

For the covering number, let κ < cov(I) and {A_{γ} : γ ∈ κ} be sets in the ideal I^{α} with the corresponding Adam’s winning strategies σ_{γ}. By induction on β ∈ α construct a sequence r⃗ ∈ ℝ^{α} such that r⃗(β) ∉ ⋃_{γ} σ_{γ}(r⃗ ↾ β). This is possible owing to the fact that the I-small sets σ_{γ}(r⃗ ↾ β) : γ ∈ κ do not cover the whole real line. By the construction, r⃗ ∈ ℝ^{α} \ ⋃_{γ} A_{γ} as desired.

For the additivity number, let κ < add(I) and {A_{γ} : γ ∈ κ} be sets in the ideal I^{α} with the corresponding Adam’s winning strategies σ_{γ}. Let σ be Adam’s strategy such that for every sequence r⃗ ∈ ℝ^{<α} the set σ(r⃗) is some I-small set covering the I-small set ⋃_{γ} σ_{γ}(r⃗). Clearly, σ is Adam’s winning strategy for the set ⋃_{γ} A_{γ}, so ⋃_{γ} A_{γ} ∈ I^{α} as desired.

The previous lemma leaves open the question of the relationship of the cardinals non(I) and non(I^{α}). Clearly non(I) ≤ non(I^{α}), but is it possible that non(I) is strictly smaller? This is an interesting question and some unexpected methods and applications come into play.

Lemma 4.11. • non(ctble) = ℵ_{1} and non(ctble^{α}) < ℵ_{4} for every nonzero countable ordinal α.

• Consistently non(ctble^{ω}) = ℵ_{2}.

Shelah recently proved a form of club guessing on ℵ_{2} that makes it possible to lower the cardinal ℵ_{4} to the optimal ℵ_{3}.

Proof. The key tool in the argument for the first item is the following

Fact 4.12. (Shelah) [25] III 2.14, [18] 2.1.2. For all regular cardinals κ, λ with λ > κ^{+} there is a sequence ⟨C_{δ} : δ ∈ λ⟩ and a stationary set S ⊂ λ consisting of ordinals of cofinality κ so that

1. for all ordinals δ ∈ λ C_{δ} ⊂ δ is a set of ordertype ≤ κ and for δ ∈ S the set C_{δ} is actually cofinal in δ

2. for every ordinal δ ∈ λ and every γ ∈ C_{δ} there is β ∈ γ such that C_{δ} ∩γ = C_{β}

3. for every closed unbounded set E ⊂ λ there is an ordinal δ ∈ S such that C_{δ} ⊂ E.

Fix a nonzero countable ordinal α. The proof of the first item in the lemma divides into two branches according to the value of 𝔠. If 𝔠 ≤ ℵ_{2} then we are done since |ℝ^{α}| = 𝔠 < ℵ_{4}. If 𝔠 ≥ ℵ_{3} then fix a collection {r_{δ} : δ ∈ ω_{3}} of distinct reals and the club guessing sequence ⟨C_{δ} : δ ∈ ω_{3}⟩ with the set S ⊂ ω_{3} from Claim 4.12. For each ordinal δ ∈ S define the α-sequence r⃗_{δ} of reals to include the real numbers indexed by the first α many elements of the set C_{δ}. The proof will be complete once I show that the set {r⃗_{δ} : δ ∈ S} is ctble^{α}-positive.

For the ease of notation regard the sequences r⃗_{δ} as indexed by the first α many elements of the set C_{δ}. Let σ be Adam’s strategy; I must produce an ordinal δ ∈ S such that the sequence r⃗_{δ} is a valid counterplay against that strategy. To that end, fix a continuous tower ⟨M_{δ} : δ ∈ ω_{3}⟩ of elementary submodels of large enough structure, all of them of size ℵ_{2} and containing ω_{2} as a subset. Let E = {δ ∈ ω_{3} : M_{δ} ∩ ω_{3} = δ}. The set E is closed unbounded, so there must be an ordinal δ ∈ S such that C_{δ} ⊂ E. Then r⃗_{δ} is the sought sequence. Indeed, for every ordinal γ among the first α elements of the set C_{δ} the set C_{δ} ∩ γ is in the model M_{γ} by (2) above. So even the sequence r⃗_{δ} ↾ γ and Adam’s challenge σ(r⃗_{δ} ↾ γ) at round γ are in the model M_{γ}. But that challenge is just a countable set of reals, so σ(r⃗_{δ} ↾ γ) ⊂ M_{γ} does not contain the real r_{γ} ∉ M_{γ} as desired.

For the second item consider the Baumgartner’s poset Q for adding a club with finite conditions and look into a model where the Forcing Axiom for Q– FA(Q)–holds. Suppose that X ⊂ ℝ^{ω} is a set of size ℵ_{1} and ⟨r_{β} : β ∈ ω_{1}⟩ is an enumeration of all the reals occuring in the sequences in the set X. A simple FA(Q) argument shows that there is a club E ⊂ ω_{1} such that for every sequence r⃗ ∈ X there are distinct integers i ≠ j such that the reals r⃗(i) and r⃗(j) are enumerated between the same two successive elements of the set E. Clearly, Adam will win the game G(X) if at round j he plays the countable set c_{j} = {r_{β} : β ∈ α} such that α is the least ordinal in the set E such that the set c_{j} contains all the finitely many reals on the list {r_{β} : β ∈ ω_{1}} that Eve has played so far.

Lemma 4.13.  • non(bounded) ≥ ℵ_{3} implies non(bounded) = non(bounded^{α})

for every nonzero countable ordinal α.

• Consistently non(bounded) = ℵ_{1} and non(bounded^{ω^{2}}) ≥ ℵ_{2001}.

Proof. The proof of the first item is the same as in the previous lemma, using the fact that 𝔟 = non(bounded) is a regular cardinal and choosing a modulo finite increasing unbounded sequence ⟨r_{δ} : δ ∈ 𝔟⟩. For the second item, go into a model in which 𝔡 = ℵ_{1} holds together with the statement “for every collection X of size ℵ_{2001} of infinite subsets of ω_{1} there is a club E ⊂ ω_{1} which does not contain any of them as a subset”. It is rather difficult to find such a model. [1] finds a model in which there is a collection of size ℵ_{2002} consisting of closed unbounded subsets of ω_{1} such that every uncountable subcollection has finite intersection, and this stays true in any ℵ_{1}-preserving extension. Our required theory will be realized in the extension of their model by a finite support iteration of ω_{1} many Hechler reals. An alternative approach stressing the techniques of the present paper will use an extension of a model of ZFC+LC+GCH by a countable support product of ℵ_{2002} many copies of the poset P_{T} obtained in Subsection 6.1.

Suppose X ⊂ ℝ^{ω^{2}} is a set of size ℵ_{2001}. To see how Adam wins the game G(X), fix a modulo finite increasing and cofinal sequence ⟨f_{β} : β ∈ ω_{1}⟩ in ω^{ω}. For each sequence r⃗ ∈ ℝ^{ω^{2}} let cr⃗ be the set of ordinals β ∈ ω for which there is an integer n such that β is the smallest ordinal with f_{β} modulo finite dominating all the reals {r⃗_{γ} : γ ∈ ω · n}. Let E ⊂ ω_{1} be a closed unbounded set which does not contain any of the sets cr⃗ : r⃗ ∈ X as a subset provided it is infinite. Adam will win the game G(X) by playing in each round the set of all functions in ω^{ω} modulo finite dominated by f_{β} where β is the smallest ordinal in the set E with f_{β} modulo finite dominating previous Eve’s answers. It is easy to check that if some sequence r⃗ ∈ X were a legal counterplay against this strategy, the set cr⃗ would be infinite and it would be contained in the set E, contradicting the choice of the set E.

The computation of non(I^{α}) for other ideals I seems to involve even more complex phenomena. The applications to duality theorems in Subsection 7.3 seem to make it worthwhile to look into these problems.

4.3 The case of a projectively generated ideal

I now want to show that given a provably proper definable real forcing P and a countable ordinal α, the ideal I(P_{α}) is the same as the Fubini power (I(P))^{α} as far as universally Baire sets are concerned. The difficulty at this stage is that the game defining the Fubini powers is too complex to be determined–Adam’s moves have a rather arbitrary complexity. It is necessary to approximate the transifinite game using a game with real moves.

In this section suppose that the universally Baire part of the ideal I(P) is equal to its projective part, which is some projectively generated projectively defined ideal I. The game defining the ideal I^{α} can then be assumed to have real moves–Adam will just play codes for projective sets in the ideal I. The following dichotomy is critical:

Lemma 4.14. (ZFC+LC) For every countable ordinal α and every universally Baire set A ⊂ ℝ^{α},

• either A ∈ I^{α} and then P_{α} ⊩ r⃗_{gen} ∉ Ȧ

• or A ∉ I^{α} and then A has a Borel I(P)-perfect subset B. Clearly B ⊩ r⃗_{gen} ∈ Ȧ.

Proof. Fix the ordinal α and the universally Baire set A ⊂ ℝ^{α}. By Fact C.8, the game G(A) is determined and moreover, one of the players has a universally Baire winning strategy.

Suppose first that Adam has a universally Baire winning strategy σ. Then P_{α} forces that the generic sequence r⃗_{gen} ∈ ℝ^{α} is a legal counterplay against the strategy in the extension. The reason is that for every β ∈ α the set σ(r⃗_{gen} ↾ β) is I(P)-small in the model V [r⃗_{gen} ↾ β] and the real r⃗_{gen}(β) falls out of all such sets. Now by the universally Baire absoluteness, the strategy σ must be winning in the game G(A) even in the extension and therefore r⃗_{gen} is forced to fall out of the set Ȧ.

Now suppose that Eve has a universally Baire winning strategy σ. By induction on β ∈ α+1 construct universally Baire I(P)-perfect sets A_{β} ⊂ ℝ^{β} and universally Baire functions f_{β} with domain A_{β} so that:

• for every sequence r⃗ ∈ A_{β} the functional value f_{β}(r⃗) is a partial play of the game G(A) of length β in which Eve follows the strategy σ and produces the sequence r⃗

• for γ ∈ β ∈ α + 1 it is the case that A_{γ} = A_{β} ↾ γ and for every sequence r⃗ ∈ A_{β} f_{β}(r⃗) ↾ γ = f_{γ}(r⃗ ↾ γ).

In the end, A_{α} is a universally Baire I(P)-perfect subset of ℝ^{α} which is also a subset of A, because for every sequence in it the function f_{α} indicates a play respecting the winning strategy σ which resulted in that sequence. Then use Corollary 4.6 to find a Borel I-perfect set B ⊂ A_{α}, completing the proof of the lemma.

The induction process is quite simple. At the limit stages β, let A_{β} = {r⃗ ∈ ℝ^{β} : ∀γ ∈ β r⃗ ↾ γ ∈ A_{γ}}, and for every sequence r⃗ ∈ A_{β} let f_{β}(r⃗) = ⋃_{γ∈β} f_{γ}(r⃗ ↾ γ). At the successor stages, let A_{β+1} = {r^{⌢}s : r⃗ ∈ A_{β} and for some partial play of the game G(A) of length β+1 respecting the strategy σ and extending the play f_{β}(r⃗), the real s is Eve’s last move}. The only nontrivial step in the verification of the I(P)-perfectness of the set A_{β+1} is to show that every sequence r⃗ ∈ A_{β} branches into I(P)-positively many reals. Well, if the set X of the sons of this sequence were I(P)-small, then as a universally Baire I(P)-small set it would have a projective small superset, which could be used as Adam’s move after f_{β}(r⃗) has been played, forcing Eve to play a real s out of X, contradicting the definition of the set X. This is the only place where the simplifying assumption on the ideal I(P) is used. To get the function f_{β+1}, use the universally Baire uniformization to choose a play f_{β+1}(r^{⌢}s) extending f_{β}(r⃗) which respect the strategy σ and results in the sequence r^{⌢}s, this for every r^{⌢}s ∈ A_{β+1}.

The dichotomy has several corollaries important in the applications:

Corollary 4.15. The universally Baire parts of the ideals I^{α}, (I(P))^{α} and I(P_{α}) coincide.

Proof. Suppose that A ⊂ ℝ^{α} is a universally Baire set. The equivalence A ∈ I^{α} if and only if A ∈ I(P_{α}) is just the statement of the dichotomy. If A ∈ I^{α} then certainly A ∈ (I(P))^{α} because in the latter game Adam has strictly more possibilities. On the other hand, if A ∈ (I(P))^{α} then A certainly does not have a Borel I-perfect subset and so by the dichotomy A ∈ I^{α}.

Corollary 4.16. Suppose that I is a homogeneous ideal. If {A_{γ} : γ ∈ λ} is a collection of < cov(I) many projective subsets of ℝ^{α}, each without a Borel I-perfect subset, then even their union does not contain a Borel I-perfect subset.

Proof. Such sets A_{γ} must be I^{α}-small by the dichotomy. The Corollary then follows by an argument parallel to the first part of Lemma 4.10.

Corollary 4.17. If I is homogeneous then so is I^{α} for every countable ordinal α.

Proof. Let α be a nonzero countable ordinal and C ⊂ ℝ^{α} a Borel I^{α}-positive set. Thinning out the set C if necessary we may assume that in fact C is I-perfect– Lemma 4.14. For each ordinal β ∈ α and each sequence s⃗ ∈ C ↾ β choose a function fs⃗ : ℝ → {r ∈ ℝ : s^{⌢}r ∈ C ↾ β + 1} such that fs⃗-preimages of I-small sets are I-small. Consider the function f : ℝ^{α} → C obtained in the following way. For a sequence r⃗ ∈ ℝ^{α}, the functional value is the unique sequence s⃗ ∈ C such that for every ordinal β ∈ α s⃗(β) = f_{s⃗↾β}(r⃗(β)). It is not difficult to verify that the f-preimages of I^{α}-small sets are I^{α}-small.

Lemma 4.14 says that in the models of the form L(ℝ)(Γ) satisfying Real Determinacy, every set A ⊂ ℝ^{α} is either I^{α}-small or else it has a Borel I-perfect subset. This dichotomy can actually hold in larger models too.

Lemma 4.18. (ZFC+LC) Suppose that F is a Ramsey ultrafilter on ω and Γ is a class of universally Baire sets of reals such that L(ℝ)(Γ) ∩ P(ℝ) = Γ and the model L(ℝ)(Γ) satisfies Real Determinacy. Then in the model L(ℝ)(Γ)[F], the following holds for every countable ordinal α and every set A ⊂ ℝ^{α}: either the set is ctble^{α}-small, or it has a ctble-perfect subset.

Proof. The key fact is that the ultrafilter F is L(ℝ)(Γ)-generic for the poset Q = P(ω)\finite sets ordered by inclusion [40]. So fix a countable ordinal α and a set A ⊂ ℝ^{α} in the model L(ℝ)(Γ)[F]; there must be a name Ȧ ∈ L(ℝ)(Γ) such that A = Ȧ/F. Look at the canonical c.c.c. partial order P ∈ L(ℝ)(Γ)[F] adding a set ȧ ⊂ ω diagonalizing the filter F. Suppose a ⊂ ω is a V -generic set for this poset. By in V [a] there is an elementary embedding j : (L(ℝ)(Γ))^{V} → (L(ℝ)(Γ))^{V [a]} which does not move the ordinals. Since every set of reals in the model L(ℝ)(Γ) is Souslin, this embedding is unique and canonically given by the set ℝ^{V [a]}. Since ℝ^{V [a]} = {f(a) : f ∈ V is a Borel function} ∈ (L(ℝ)(Γ))^{V} [F][a], the embedding j belongs to the model (L(ℝ)(Γ))^{V} [F][a] and so there is a name for it in the model L(ℝ)(Γ)[F]. Consider the P -name Ḃ = {r⃗ ∈ ℝ^{α} : (L(ℝ)(Γ))^{V [ȧ]} ⊧ ȧ ⊩_{Q} r⃗ ∈ jȦ}. Since the set ȧ is forced to diagonalize the filter F, an absoluteness argument can be applied to show P ⊩ Ḃ ∩ ℝ^{V} = Ǎ. There are two cases.

• either some condition p ∈ P forces that the set Ḃ is ctble^{α}-small. In such a case, Adam has a corresponding winning strategy σ in the transfinite game G(Ḃ). Since the set Ḃ is in the model (L(ℝ)(Γ))^{V [a]} which satisfies Real Determinacy, such a strategy can be found in this model, and therefore there must be a name σ̇ for it in the model (L(ℝ)(Γ))^{V} [F]. Consider Adam’s strategy τ in the model (L(ℝ)(Γ))^{V} [F] given by τ(s⃗) = {r ∈ ℝ : some condition q ≤ p in the poset P forces ř into σ̇(s⃗)} for all sequences s⃗ ∈ ℝ^{<α}. By the c.c.c. of the forcing P, the sets τ(s⃗) are countable and therefore this is a strategy. I claim this is a winning strategy against the set A and L(ℝ)(Γ)[F] ⊧ A ∈ ctble^{α}. Indeed, fix a sequence r⃗ ∈ A. Then p ⊩ r⃗ ∈ Ḃ and since the strategy σ̇ is forced to be winning, there must be a condition q ≤ p and an ordinal β ∈ α such that q ⊩ r⃗(β̌) ∈ σ̇(r⃗ ↾ β̌). Then r⃗(β) ∈ τ(r⃗ ↾ β) and r⃗ is not a legal play against the strategy τ.

• or P forces that the set Ḃ has a Borel ctble-perfect subset. Then, given a set b ∈ F, the condition ⟨0, b⟩ ∈ P forces that the following statement to hold in (L(ℝ)(Γ))^{V [a]}: there is a set c ⊂ b such that the set {r⃗ ∈ ℝ^{α} : c ⊩_{Q} r⃗ ∈ jȦ} contains a Borel ctble-perfect subset; namely the set ȧ is such. By the elementarity of the embedding j it follows that there is such a set c ⊂ b already in the model (L(ℝ)(Γ))^{V}. By the genericity of the filter F, there must be such a set c in the filter F. Then clearly L(ℝ)(Γ)[F] ⊧ A contains a Borel ctble-perfect set.

Lemma 4.19. Suppose that κ is an inaccessible cardinal. The following dichotomy holds in the choiceless Solovay model collapsing κ to become ω_{1}: for every ordinal α ∈ κ and every set A ⊂ ℝ^{α}, either the set is ctble^{α}-small or it contains a Borel ctble-perfect subset.

A general theorem implying this lemma will be proved in a forthcoming work.

4.4 The general case

Now consider the case of an arbitrary proper definable real forcing P; so possibly the universally Baire part of the ideal I(P) can be strictly larger than its projective part. It is now necessary to replace the games defining the ideal (I(P))^{α} by the following game H(A), for a set A ⊂ ℝ^{α}: The game lasts α many rounds. At each round β ∈ α Eve starts, plays a I(P)-positive Borel set B_{β}, then Adam plays a positive Borel set C_{β} ⊂ B_{β} and finally Eve chooses a real r⃗(β) ∈ C_{β}. Eve wins if the sequence r⃗ she produced belongs to the set A. This game has the same outcome as the game G(A) for our special case of ideals of the form I(P) where P is proper:

Lemma 4.20. Let A ⊂ ℝ^{α} be a set. Adam has a winning strategy in the game G(A) if and only if he has a winning strategy in the game H(A).

Proof. Fix a set A ⊂ ℝ^{α}. Any Adam’s winning strategy σ in the game G(A) will translate into a winning strategy τ in H(A) round by round in the following way: at a given position in the play of H(A) Adam looks at the move the strategy τ dictates to him in the corresponding position of the play of G(A). It is some I(P)-small set X_{β}, and therefore whatever positive Borel I(P)-positive set B_{β} Eve may play at this stage in the game H(A), Adam can find its positive Borel subset C_{β} with empty intersection with the set X_{β}. The play then proceeds into the next round. The translation of a winning strategy τ in the game H(A) into a winning strategy σ in the game G(A) is similar. At a given position in the play of the game G(A) Adam plays the set X_{β} = ℝ \ ⋃{C_{β} : C_{β} is Adam’s answer the strategy τ dictates to him after Eve plays some set B_{β} in the corresponding position of the play of the game H(A)}. This set is I(P)-small by Lemma 2.10.

Lemma 4.21. (ZFC+LC) For every countable ordinal α and every universally Baire set A ⊂ ℝ^{α},

• either A ∈ (I(P))^{α} and then P_{α} ⊩ r⃗_{gen} ∉ Ȧ

• or A ∉ (I(P))^{α} and then A has a Borel I(P)-perfect subset B. Clearly B ⊩ r⃗_{gen} ∈ Ȧ.

The proof of this dichotomy is a repetition of the argument for Lemma 4.14 using the determinacy of the game H(A).

Corollary 4.22. The universally Baire parts of the ideals (I(P))^{α} and I(P_{α}) coincide.

Corollary 4.23. (ZFC+LC) Assume that FA(P) holds. Whenever {A_{γ} : γ ∈ ω_{1}} are projective subsets of ℝ^{α}, each without a Borel I(P)-perfect subset, then even their union has no Borel I(P)-perfect subset.

Here FA(P), the Forcing Axiom for P, is the statement asserting that for every collection X of size ℵ_{1} consisting open dense subsets of P and every condition p ∈ P there is a filter on P containing the condition p and meeting every element of X.

Proof. Such sets A_{γ} must be (I(P))^{α}-small and there will be corresponding Adam’s winning strategies {σ_{γ} : γ ∈ ω_{1}} in the games H(A_{γ}). Suppose for contradiction that the union does contain a Borel I(P)-perfect subset B. To prove the lemma, I need to produce plays H_{γ} : γ ∈ ω_{1} against the strategies σ_{γ} that will result in the same sequence r⃗ ∈ B.

Now these plays will be produced by simultaneous recursion on the round β ∈ α. To perform the recursion, suppose that the fragments H_{γ} ↾ β of the plays H_{γ} : γ ∈ ω_{1} of length β have been obtained resulting in the same β-sequence r⃗ ↾ β ∈ B ↾ β of reals. Let B_{β} be the I(P)-positive Borel set {s ∈ ℝ : (r⃗ ↾ β)^{⌢}s ∈ B ↾ β + 1}. The forcing axiom for P (or P_{I(P)}) guarantees that there is an elementary submodel M of a large enough structure containing all the relevant information such that |M| = ℵ_{1} and ω_{1} ⊂ M, and an M-generic filter G ⊂ P_{I(P)} containing the condition B_{β} [21] Lemma 38. The M-generic real s given by this filter will belong to the set B_{β}. Moreover, by a density argument applied in the model M, for each ordinal γ ∈ ω_{1} there will be sets B_{β}^{γ} and C_{β}^{γ} in the filter G such that the partial play H_{γ} ↾ β + 1 = (H_{γ} ↾ β)^{⌢}(B_{β}^{γ})^{⌢}(C_{β}^{γ})^{⌢}s respects the strategy σ_{γ}–note that s ∈ C_{β}^{γ} as C_{β}^{γ} ∈ G. The recursion will proceed with r⃗ ↾ β + 1 = (r⃗ ↾ β)^{⌢}s. At limit stages β the sequence r⃗ ↾ β is guaranteed to belong to the set B ↾ β since the set B is I-perfect.

The previous corollary has strictly stronger assumption than the corresponding statement from the previous subsection. In some situations, when the Borel part I of the ideal I(P) is well understood–like in the case of the E_{0}-forcing in Subsection 3.6–it would be better to deal with cov(I) directly as opposed to the forcing axiom for P, just as it was done in the previous section. However, if the assumption that the universally Baire part of the ideal I(P) is equal to its projective part is absent, the resulting estimates must consider the following determinacy parameter. Let θADℝ be the minimal possible θ of a transitive model of Real Determinacy and Dependent Choice containing all reals and ordinals all of whose sets are universally Baire and put 𝔡𝔢𝔱 = cof[θADℝ]^{ℵ_{0}}, the minimal size of a family X ⊂ [θADℝ]^{ℵ_{0}} such that every countable subset of θADℝ is a subset of an element of the family X. This sounds terribly complicated, but in all known models of ZFC+LC it is the case that 𝔡𝔢𝔱 = |θADℝ| < ℵ_{3}, and in all proper forcing extensions of models of ZFC+LC+CH it is even true that 𝔡𝔢𝔱 = |θADℝ| = ℵ_{1} by Fact C.5. It is a major open problem in set theory whether the value of θADℝ can be manipulated beyond the ℵ_{3} restriction.

Corollary 4.24. (ZFC+LC) Let P be a provably proper definable real forcing, and let I be the projective (or Borel) part of the ideal I(P). Suppose that I is a homogeneous ideal. Suppose that 𝔡𝔢𝔱 < cov(I). For every countable ordinal α, whenever {A_{γ} : γ ∈ κ} are < cov(I) many projective subsets of ℝ^{α}, each of them without a Borel I-perfect subset, then even their union has no Borel I-perfect subset.

Proof. Fix a transitive model M of Real Determinacy and Dependent Choice containing all the reals such that θADℝ = θ^{M}. First, a small observation:

Claim 4.25. Every set A ⊂ ℝ in the model M can be written as a union of ≤ 𝔡𝔢𝔱-many Borel sets.

Proof. By Fact C.9, there is a tree T ⊂ (λ × ω)^{<ω} in the model M for some ordinal λ ∈ θADℝ such that A = p[T]. Let {x_{β} : β ∈ 𝔡𝔢𝔱} be a collection of countable subsets of λ such that every countable subset of x is a subset of one of them. For every ordinal β ∈ κ let A_{β} = p[T ↾ (x × ω)^{<ω}]. Clearly A = ⋃_{β} A_{β} and each A_{β} is an analytic set. Every analytic set is a union of ℵ_{1} many Borel sets. The claim follows.

Corollary 4.26. Suppose 𝔡𝔢𝔱 < cov(I). No Borel I-positive set is covered by a union of less than cov(I) many sets in the model M which do not contain any Borel I-positive subsets.

Proof. Let {C_{γ} : γ ∈ λ}, λ < cov(I) be a collection of M-sets of reals, neither of which contains a Borel I-positive subset. By Claim 4.25, each of them is a union of ≤ κ many Borel sets, and these Borel sets must be I-small. Since cov(I) > max(λ, 𝔡𝔢𝔱) and I is a homogeneous ideal, no I-positive Borel set can be covered by these 𝔡𝔢𝔱 · λ many Borel I-small sets.

Now let α be a countable ordinal. The sets {A_{γ} : γ ∈ κ} must be (I(P))^{α}- small by Lemma 4.21, and therefore Adam has winning strategies σ_{γ} in the games H(A_{γ}) and each of these strategies is in the model M. Suppose for contradiction that the union ⋃_{γ} A_{γ} contains some Borel I-perfect subset B. The proof will be completed with a construction of plays H_{γ} of the respective games following the strategies σ_{γ} such that all of them result in the same sequence r⃗ ∈ B. As before, these plays will be constructed by simultanous induction on the round β ∈ α. Suppose that the fragments H_{γ} ↾ β : γ ∈ λ have been constructed, resulting in the same sequence r⃗ ↾ β which is in the set B ↾ β. For each ordinal γ ∈ λ let C_{γ} = {s ∈ ℝ : for no Eve’s move in the next round following the fragment H_{γ} does the strategy σ_{γ} respond with a set containing the real s}. All of these sets are in the model M and they do not contain any Borel I-positive subset, because such a set could be used as Eve’s move. By Corollary 4.26, there must be a real s in the Borel I-positive set {t ∈ ℝ : (r⃗ ↾ β)^{⌢}t ∈ B ↾ β +1} which falls out of all the sets C_{γ} : γ ∈ λ. By the definition of the sets C_{γ} each of the fragments H_{γ} ↾ β has an extension H_{γ} ↾ β + 1 which observes the strategy σ_{γ} and where the last Eve’s move was exactly the real s. The induction continues after setting r⃗ ↾ β + 1 = (r⃗ ↾ β)^{⌢}s.

The induction keeps going at the limit stages since the set B is I-perfect.

5 Illfounded iterations

Let P be a proper definable real forcing, and let ⟨L, ≤⟩ be a linear ordering. Is it possible to iterate the poset P along the ordering L? This would mean finding a proper partial order P_{L} adding a function r⃗_{gen} from L to ℝ such that for every element i ∈ L, the real r⃗_{gen}(i) is P -generic over the model V [r⃗_{gen} ↾< i], where < i = {j ∈ L : j < i}. The previous sections show how to iterate the forcing P along an ordinal. In the case of non-wellfounded ordering L the arguments based on transfinite induction and determinacy of transfinite games become useless and it is even difficult to conceive a suitable definition of the ordering P_{L}. In certain very specific cases a solution has been known:

Fact 5.1. [9] The Sacks forcing can be iterated along any linear order.

and in other quite specific cases it is possible to show that a suitable iteration does not exist:

Fact 5.2. [4] (Hjorth) It is impossible to iterate any forcing adding a dominating real along any illfounded linear ordering.

This section contains a solution for a reasonably wide class of partial orders. I will need the following definition:

Definition 5.3. (Shelah) A partial order P is strongly proper if for every countable elementary submodel M of a large enough structure, every condition p ∈ P ∩ M and every collection {D_{n} : n ∈ ω} of dense subsets of P ∩ M there is a condition q ≤ p such that all the sets D_{n} are predense below q.

Thus every strongly proper forcing is proper–just consider the collection {D ∩M : D ∈ M is an open dense set}. Also, great many definable real forcings are strongly proper: all forcings of the form P_{I} where I is a σ-ideal σ-generated by closed sets, the E_{0} forcing of Subsection 3.6, the full support product of countably many Sacks reals and many more.

Theorem 5.4. (ZFC+LC) Every strongly proper definable real forcing can be iterated along an arbitrary linear ordering.

While the iteration is in no way uniquely given by the linear order and the iterand, I will obtain an iteration which has a defensible claim to canonicity. It is strongly proper, in the case of a wellfounded linear order it gives the usual countable support iteration, and with the Sacks forcing it gives the iteration discovered by Kanovei in [9] using purely combinatorial methods.

5.1 Strongly proper forcings

I will first want to establish the basic properties of real strongly proper forcings, in particular it will be useful to characterize those ideals I for which the poset P_{I} is strongly proper. First, a bit of notation. Suppose that I is a σ-ideal on the reals and M is a countable elementary submodel of a large enough structure.

The poset P_{I} ∩ M can be understood as a countable approximation of P_{I}. If g is a V -generic filter on P_{I} ∩ M, look at the real ṙ_{gen}/g. By Lemma 2.1 applied in the model M and analytic absoluteness between the models M[g] and V [g], we have g = {B ∈ P_{I} ∩ M : ṙ_{gen}/g ∈ B}, and ṙ_{gen}/g is the unique real with this property. By an abuse of notation, I will denote the P_{I} ∩ M-name for the real ṙ_{gen}/g by ṙ_{gen} again.

Lemma 5.5. Let I be a σ-ideal on the reals. The following are equivalent:

• P_{I} is strongly proper

• for every countable elementary submodel M of a large enough structure, P_{I} ∩M ⊩ ṙ_{gen} does not belong to any Borel I-small set coded in the ground model.

The same equivalence holds even for universally Baire sets from the ideal I(P_{I}).

Proof. Suppose that the first item holds, and suppose for contradiction that the second item fails. Thus there is a model M, a condition p ∈ P_{I} ∩ M and a Borel set B ∈ I such that in the forcing P_{I} ∩ M p ⊩ ṙ_{gen} ∈ Ḃ. Then the set X = {g ⊂ P_{I} ∩ M : g is a filter, p ∈ g and ṙ_{gen}/g ∈ B} must be comeager in the set {g ⊂ P_{I} ∩ M : g is a filter and p ∈ g} with the natural topology. Let {D_{n} : n ∈ ω} be the collection of open dense subsets of P_{I} ∩ M witnessing the comeagerness, and let q ≤ p be the condition from the definition of strong properness. Clearly, in the poset P_{I} the condition q forces Ġ ∩ M̌ ∈ Ẋ and so ṙ_{gen} ∈ Ḃ, contradicting the fact that B ∈ I.

Suppose now that the second item holds. To verify the strong properness of the poset P_{I}, let M be a countable elementary submodel of a large enough structure, let p ∈ P_{I} be a condition and let {D_{n} : n ∈ ω} be dense sets in the poset P_{I} ∩M. Note that p as well as the elements of D_{n}’s are Borel sets of reals. It is enough to verify that the Borel set q = p ∩ ⋂_{n} ⋃ D_{n} is I-positive, because then it is the condition required in the definition of strong properness. But the condition p in the poset P_{I} ∩ M forces ṙ_{gen} ∈ q̇ by the density of the sets D_{n}. The second item now applies to show that q ∉ I as desired.

There are some trivial consequences:

Corollary 5.6. The only c.c.c. strongly proper real forcing is the Cohen forcing. No strongly proper real forcing can add a dominating real.

Proof. For the first sentence, suppose that I is a σ-ideal and P_{I} is a c.c.c. strongly proper notion of forcing. The lemma shows that given a countable elementary submodel M of a large enough structure, the countable poset P_{I} ∩M adds a real that falls out of all Borel I-small sets of reals. By the c.c.c., such a real must be P_{I}-generic. Thus P_{I} regularly embeds into P_{I} ∩ M which is forcing-equivalent to the Cohen forcing.

The second sentence has a similar proof, noting that the Cohen forcing does not add a dominating real.

and there are some less trivial consequences:

Corollary 5.7. (ZFC+LC) Suppose that I is a projectively definable ideal such that P_{I} is strongly proper. Then P_{I} is strongly proper in all forcing extensions.

Proof. Choose an arbitrary suitably definable coding of elements of the poset P_{I} by reals and view the poset alternately as consisting of reals or Borel I-positive sets. Consider the set Mod = {M ∈ [ℝ]^{ℵ_{0}} : for every collection {O_{n} : n ∈ ω} of open dense subsets of P_{I} ∩M and every Borel I-small set B the set ⋂_{n} ⋃ O_{n} \B is nonempty}. Essentially by the definitions, the forcing P_{I} is strongly proper if and only if the set Mod contains a closed unbounded subset. Now the point is that the set Mod is in the model L(ℝ) and Real Determinacy implies that the club filter on [ℝ]^{ℵ_{0}} is an ultrafilter. This means that in every model L(ℝ)(Γ) satisfying Real Determinacy, all of whose sets of reals are universally Baire, there is a set A ⊂ ℝ such that L(ℝ)[A] ⊧ A codes a closed unbounded subset of the set Mod. By the universally Baire absoluteness, this holds in every forcing extension, and therefore P_{I} is a strongly proper poset in every forcing extension of the universe.

For the benefit of the reader and for later use let me show how Real Determinacy implies that the club filter on [ℝ]^{ℵ_{0}} is an ultrafilter. Let C ⊂ [ℝ]^{ℵ_{0}} be an arbitrary set and consider a real game in which players I and II alternately produce reals r_{n} : n ∈ ω and player I wins if {r_{n} : n ∈ ω} ∈ C. By the Real Determinacy, either player I has a winning strategy σ : ℝ^{<ω} → ℝ–and then the closed unbounded set of all countable subsets of the reals closed under the function is a subset of C–or player II has a winning strategy, in which case the same argument gives a club disjoint from the set C.

The method of proof here is more important than the fact itself. It shows that the formula “P_{I} is strongly proper” is a statement of the model L(ℝ)(Γ) for a suitable absolute enough class Γ.

The proof of Lemma 2.3 also immediately gives

Corollary 5.8. Whenever I is a σ-ideal σ-generated by closed sets then the forcing P_{I} is strongly proper.

5.2 Countable length

For this and the next subsection fix a strongly proper definable real forcing P and let I = I(P) so that P = P_{I}. The present subsection computes an iteration P_{L} of the poset P along a countable linear ordering L. The construction of the iteration proceeds along the following course.

1. define a suitable ideal I^{L} on the set ℝ^{L}; the poset P_{I^{L}} = P_{L} will be the desired iteration.

2. prove that for every initial segment K ⊂ L the ideal I^{L} is naturally isomorphic to the Fubini product of the ideals I^{K} and I^{K\L}. As a consequence, the poset P_{K} adding the first K many reals can be naturally viewed as a regular suborder of the poset P_{L}.

3. prove that the poset P_{L} is nontrivial and proper

4. prove that the poset P_{L} has the properties expected from an iteration.

Using C.8 find a model L(ℝ)(Γ) of Real Determinacy and Dependent Choice all of whose sets of reals are universally Baire. The model is correct about the membership in the ideal I and I will abuse the notation by confusing I and I ∩ M. The ideal I^{L} is defined inside this model; the same definition in the ZFC universe gives an improper ideal–Example 5.15. Until further notice work within this model.

Let I_{0}^{L} be the ideal σ-generated by sets A_{g} where for some element i ∈ L g is a function from ℝ^{<i} into the Borel I-small sets and A_{g} = {r⃗ ∈ ℝ^{L} : r⃗(i) ∈ g(r⃗ ↾< i)}. The partial order of I_{0}^{L}-positive Borel sets with inclusion seems to be a natural candidate for the iteration of P_{I} along the ordering L. However, it is not clear how to prove its properness, and the obstacle is the apparent impossibility to argue that every I_{0}^{L}-positive set has a Borel positive subset. Lemma 2.13 (1) and Lemma 2.12 show that this is an essential difficulty. It is necessary to correct the ideal I_{0}^{L}–to find a slightly bigger ideal I^{L} ⊃ I_{0}^{L} which satisfies this Borel density property. Just let I^{L} be the smallest ordinal-covering σ-ideal including I_{0}^{L} as a subset using 2.13. It is not clear whether in general P_{I^{L}} = P_{I_{0}^{L}} or not. The equality does hold in the case of a wellfounded ordering L. This is proved using a real game along L as in the previous section. In the illfounded case this argument is useless and I have no information.

Lemma 5.9. For every initial segment K ⊂ L, the ideal I^{L} is naturally isomorphic to the Fubini product of the ideals I^{K} and I^{L\K}.

Proof. Fix an initial segment K ⊂ L. The lemma says that a set C ⊂ ℝ^{L} is I^{L}-positive if and only if {s⃗ ∈ ℝ^{K} : {t⃗ ∈ ℝ^{L\K} : s^{⌢}t⃗ ∈ C} ∉ I^{L\K}} ∉ I^{K}. For a sequence s⃗ ∈ ℝ^{K} write C(s⃗) = {t⃗ ∈ ℝ^{L\K} : s^{⌢}t⃗ ∈ C} and write C ↾↾ K = {s⃗ ∈ ℝ^{K} : C(s⃗) ∉ I^{L\K}}. The following two claims complete the proof of the lemma.

Claim 5.10. If C ↾↾ K ∉ I^{K} then C ∉ I^{L}.

Proof. Let J be the collection of all sets A ⊂ ℝ^{L} such that A ↾↾ K ∈ I^{K}. It is not difficult to see that J is a σ-ideal. The claim will be proved if I show that I^{L} ⊂ J and this in turn will follow from I_{0}^{L} ⊂ J and the ordinal covering of J.

To see that I_{0}^{L} ⊂ J choose a function g : ℝ^{<i} → I and suppose for contradiction that A_{g} ∉ J. If i ∈ K just use the positivity of A_{g} ↾↾ K to pick s⃗ ∈ A_{g} ↾↾ K such that s⃗(i) ∉ g(s⃗ ↾ i) and then choose an arbitrary sequence t⃗ ∈ A_{g}(s⃗). Clearly s^{⌢}t⃗ should both belong to and fall out of the set A_{g}, contradiction. If i ∉ K then choose an arbitrary sequence s⃗ ∈ C ↾↾ K and then use the positivity of the set C(s⃗) to find a sequence t⃗ in this set such that t⃗(i) ∉ g(s^{⌢}t⃗ ↾< i). This leads to the same contradiction as before.

For the ordinal covering of the ideal J let C ⊂ ℝ^{L} be a J-positive set and let f : C → [λ]^{ℵ_{0}} be a function. For every sequence s⃗ ∈ C ↾↾ K use the positivity of the set C(s⃗) to conclude that there must be countable set x ⊂ λ such that the set {t⃗ ∈ C(s⃗) : f(s^{⌢}t⃗) ⊂ x} is I^{L\K}-positive. Use the uniformization C.9 to find a function h : C ↾↾ K → [λ]^{ℵ_{0}} picking one such set x for every sequence s⃗. The positivity of the set C ↾↾ K implies that there will be a countable set x ⊂ λ such that h^{−1}[x]^{ℵ_{0}} is I^{K}-positive. By inspection, the set D = f^{−1}[x]^{ℵ_{0}} has D ↾↾ K ∉ I^{K} and so is J-positive as required.

Claim 5.11. If C ↾↾ K ∈ I^{K} then C ∈ I^{L}.

Proof. Define a collection J of subsets of ℝ^{K} by A ∈ J if there is no I^{L}-positive set C ⊂ ℝ^{L} such that A = C ↾↾ K. The claim will follow from the equality I^{K} = J, and this equality will in turn follow once it is shown that J is an ordinally covering σ-ideal with I_{0}^{K} ⊂ J ⊆ I^{K}.

It is not difficult to see that J is indeed a σ-ideal including the ideal I_{0}^{K}. If A ⊂ ℝ^{K} is an I^{K}-positive set then the set C = {r⃗ ∈ ℝ^{L} : r⃗ ↾ K ∈ A} is I^{L}-positive with C ↾↾ K = A. This is true by the previous claim and the fact that I^{L\K} is a proper ideal as shown in Claim 5.14. Thus J ⊂ I^{K}. And finally, suppose that A ⊂ ℝ^{K} is an J-positive set and f : A → [λ]^{ℵ_{0}} is a function. Choose an I^{L}-positive set C ⊂ ℝ^{L} such that A = C ↾↾ K and let h : C → [λ]^{ℵ_{0}} be a function defined by h(r⃗) = f(r⃗ ↾ K). Let x ⊂ λ be a countable set such that the set D = h^{−1}[x]^{ℵ_{0}} is I^{L}-positive. Clearly the set B = D ↾↾ K is an J-positive subset of f^{−1}[x]^{ℵ_{0}}, showing that the ideal J is ordinal covering. The claim follows.

The previous lemma says exactly that the partial order Q_{K} = P(ℝ^{K}) \ I^{K} with inclusion is naturally regularly embedded into Q_{L} = P(ℝ^{L}) \ I^{L}. The embedding map i_{K} : Q_{K} → Q_{L} is defined by i_{K}(B) = {r⃗ ∈ ℝ^{L} : r⃗ ↾ K ∈ B} and the associated projection map is the restriction ↾↾. Since the posets P_{L} and P_{K} are dense in the orders Q_{K} and Q_{L} respectively by Lemma 2.13, the embedding i_{K} maps P_{K} regularly into P_{L}. The reader should notice that the projection map can send a Borel set into a set of much higher complexity.

Move out of the model L(ℝ)(Γ). The following three lemmas have nearly identical proofs. The first shows that in some sense the ideal I^{L} is as small as possible. The other two show that P_{L} is a nontrivial proper notion of forcing.

Lemma 5.12. Let Q be an arbitrary proper iteration of the poset P_{I} along the linear order L adding a generic sequence r⃗_{gen}. For every set A ∈ I^{L}, Q ⊩ r⃗_{gen} ∉ Ȧ.

Proof. It is just enough to show that I^{L} ⊂ I(Q), and by the minimal choice of the ideal I^{L} it is enough to show that I_{0}^{L} ⊂ I(Q). Well, choose an arbitrary function g ∈ L(ℝ)(Γ), g : ℝ^{<i} → I. Look at the generic sequence r⃗_{gen}. The real r⃗_{gen}(i) is V [r⃗_{gen} ↾< i]-generic for the poset P_{I} and therefore must fall out of the I-small set g(r⃗_{gen} ↾ i) ∈ V [r⃗_{gen} ↾< i]. Thus r⃗_{gen} ∉ A_{g}.

Corollary 5.13. For a one-element ordering L, I = I^{L}.

Proof. Just apply the previous lemma to the partial ordering P = P_{I}.

Lemma 5.14. The ideal I^{L} is proper, that is ℝ^{L} ∉ I^{L}.

Proof. Let R_{L} be the finite support product of L many copies of the forcing P_{I}. This poset adds a generic sequence r⃗_{gen} ∈ ℝ^{L}; for every initial segment K ⊂ L the sequence r⃗_{gen} ↾ K is added by the poset R_{K} ⊂ R_{L}. Let M be a countable elementary submodel of a large enough structure containing all the necessary information. The poset R_{L} ∩ M is countable and it still adds an L-sequence of reals that I will abuse the notation to call r⃗_{gen} again. I will show that for every set A ∈ Γ ∩ I^{L}, R_{L} ∩ M ⊩ r⃗_{gen} ∉ Ȧ. By the universally Baire absoluteness this will mean that A ≠ ℝ^{L} as desired.

It is enough to verify that I^{L} ⊂ I(R_{L} ∩ M). Now the forcing R_{L} ∩ M is countable, therefore proper and by Lemma 2.12 the ideal I(R_{L} ∩ M) is Γordinal covering. So by the minimal choice of the ideal I^{L} it is just necessary to show that I_{0}^{L} ⊂ I(R_{L} ∩ M). Well, choose a function f : ℝ^{<i} →the Borel sets in the ideal I where f ∈ L(ℝ)(Γ) and i ∈ L. It should be the case that R_{L} ∩ M ⊩ r⃗_{gen}(i) ∉ f(r⃗_{gen} ↾< i).

To see this, consider the set Mod = {N ∈ ℝ^{ℵ_{0}} : for every countable collection {O_{n} : n ∈ ω} of open dense subsets of the poset P_{I} ∩ N and every Borel I-small set B the set B ∩ ⋂_{n} ⋃ O_{n} is nonempty}. This set is in the model L(ℝ)(Γ) and it contains a closed unbounded subset. Use Real Determinacy to find a function σ : ℝ^{<ω} → ℝ in the model L(ℝ)(Γ) such that all of its closure points are in the set Mod–see the proof of Corollary 5.7. Such a function σ can be found in the model M. Now let r⃗_{gen} ∈ ℝ^{L} be a V -generic sequence for the poset R_{L} ∩ M. In the model V [r⃗_{gen} ↾< i] the model M is still closed under the function σ by the universally Baire absoluteness, and therefore the real r⃗_{gen}(i), which is V [r⃗_{gen} ↾< i]-generic for P_{I} ∩ M, must fall out of all I-small sets coded in the model V [r⃗_{gen} ↾< i], in particular from the set f(r⃗_{gen} ↾< i) as desired.

The previous claim uses heavily the fact that the ideal I_{0}^{L} was defined inside a model with very strong absoluteness properties. This is a necessary concern in the case of illfounded ordering L. The following example shows that the ideal ctble_{0}^{ω^{∗}} would be improper if its definition were carried out in a ZFC universe, where ω^{∗} is the inverted ω ordering. In the wellfounded case this concern is irrelevant.

Example 5.15. (ZFC) There is a function f : ℝ^{ω} → ℝ such that for every sequence r⃗ ∈ ℝ^{ω} there is an integer i such that r⃗(i) = f(r⃗ \ i + 1). Here r⃗ \ i + 1 is the sequence s⃗ ∈ ℝ^{ω} defined by s⃗(j) = r⃗(j + i + 1).

Proof. Let E be the equivalence on ℝ^{ω} defined by r⃗Es⃗ if there are integers i, j such that r⃗ \ i = s⃗ \ j. Let A ⊂ ℝ^{ω} be a set meeting each equivalence class in exactly one sequence, and if this is an equivalence class of eventually periodic sequences then this sequence is outright periodic. Choose f : ℝ^{ω} → ℝ so that f(r⃗\i+1) = r⃗(i) if r⃗ ∈ A and i ∈ ω; the other values are irrelevant. The various requirements on the function f do not clash owing to the choice of the set A. It is not difficult to check that the function f has the required properties.

Lemma 5.16. The poset P_{L} is strongly proper.

Proof. First, let me gather some relevant facts. The poset is non-trivial by the previous lemma, and so it adds a generic sequence r⃗_{gen} ∈ ℝ^{L} of reals. Lemma 5.9 and the remarks following it say that for an arbitrary initial segment K ⊂ L the sequence r⃗_{gen} ↾ K is added by the poset P_{K} which is naturally embedded into P_{L}. Let Mod = {N ∈ [ℝ]^{ℵ_{0}} : for every countable linear order K ∈ N with first element i ∈ K it is the case that P_{{i}} ∩ N is completely embedded in P_{K} ∩ N; moreover, for every countable collection {O_{n} : n ∈ ω} of open dense subsets of the poset P_{I} and every I-small Borel set B the set A ∩ ⋂_{n} ⋃ O_{n} is nonempty}. The set Mod contains a closed unbounded set, and it belongs to the model L(ℝ)(Γ) of Real Determinacy. The Determinacy gives a function σ : ℝ^{<ω} → ℝ in the model L(ℝ)(Γ) such that every countable closure point of it belongs to the set Mod–see the proof of Corollary 5.7. Note that the function σ is universally Baire.

Now let M be a countable elementary submodel of a large enough structure containing all the necessary information, in particular σ ∈ M. Let A ∈ L(ℝ)(Γ) be an I-small set. By Lemma 5.5, I must prove that P_{L} ∩ M ⊩ r⃗_{gen} ∉ Ȧ. As in the previous lemmas, it is enough to verify this for the sets A of the form {r⃗ ∈ ℝ^{L} : r⃗(i) ∈ f(r⃗ ↾< i)} where i ∈ L and f ∈ L(ℝ)(Γ) is a function from ℝ^{<i} to the Borel sets in the ideal I. So fix such i ∈ L and f, and write K =< i = {j ∈ L : j < i}. Let G_{L} ⊂ P_{L} ∩ M be a V -generic filter and let r⃗_{gen} ∈ R^{L} be the corresponding generic sequence for the poset P_{L} ∩ M, thus {r⃗_{gen}} = ⋂ G_{L}. Write G_{K} = {B ∈ P_{K} ∩ M : {r⃗ ∈ ℝ^{L} : r⃗ ↾ K ∈ B} ∈ G_{L}}. It is clearly the case that G_{K} ⊂ P_{K} ∩M is a V -generic filter and {r⃗_{gen} ↾ K} = ⋂ G_{K}. Let X = {g(r⃗_{gen} ↾ K) : g ∈ M is a Borel function from ℝ^{K} to ℝ}.

Claim 5.17. X ∈ Mod–of course here the set Mod is evaluated in the model V [r⃗_{gen} ↾ K].

Proof. It is enough to show that the set X is closed under the function σ. Well, suppose ⟨r_{m} : m ∈ n⟩ is a finite sequence of reals in the set X. This means that there must be Borel functions g_{m} : B → ℝ in the model M for some Borel I_{K}-positive set B ⊂ ℝ^{K} in the filter G_{K} such that r_{m} = g_{m}(r⃗_{gen} ↾ K) for all numbers m ∈ n. Let h : B → ℝ be a function defined by h(r) = σ(g_{m}(r) : m ∈ n). The function h belongs to the model M and by Lemma 2.13(3), the ordinal covering of the ideal I_{K} and the genericity of the filter G_{K} there must be a set C ⊂ B in the filter G_{K} such that the function h ↾ C is a Borel function. By the universally Baire absoluteness between the models V and V [G_{K}] it must be the case that σ(r_{m} : m ∈ n) = (h ↾ C)(ṙ_{gen} ↾ K) ∈ X.

Claim 5.18. The sequence r⃗_{gen} ↾ L \ K is V [r⃗_{gen} ↾ K]-generic for the poset P_{L\K} ∩ X.

Proof. Back to V. Suppose B ∈ P_{L} ∩M is a condition and Ḋ is a P_{K} ∩M-name for a dense subset of Ṗ_{L\K} ∩ Ẋ. There is a set B_{0} ∈ P_{K} ∩ M such that for every sequence s⃗ ∈ B_{0} the set {r⃗ ∈ P_{L\K} : s^{⌢}r⃗ ∈ B} is P_{L\K}-positive. This follows from Lemma 5.9 and the ordinal covering of the ideal I_{K}. Let g_{0} : B_{0} → P_{L\K} be defined by g_{0}(s⃗) = {r⃗ ∈ P_{L\K} : s^{⌢}r⃗ ∈ B}. Thus g_{0} is a function in the model M which is in a suitable sense Borel and B_{0} ⊩ ġ_{0}(r⃗_{gen} ↾ K) ∈ Ẋ ∩ P_{L\K}.

Let B_{1} ⊂ B_{0} be a condition in the poset P_{K} ∩ M and g_{1} : B_{1} → P_{L\K} be a Borel function in the model M such that B_{1} ⊩ ġ_{1}(r⃗_{gen} ↾ K) ⊂ ġ_{0}(r⃗_{gen} ↾ K) is a condition in the set Ḋ. The set {s⃗ ∈ B_{1} : g_{1}(s⃗) ⊄ g_{0}(s⃗)} ∈ M must be I_{K}-small, because if it were positive, any Borel positive subset of it in the poset P_{K} ∩ M would force ġ_{1}(r⃗_{gen} ↾ K) ⊄ ġ_{0}(r⃗_{gen}(↾ K), contradicting the choice of the condition B_{1}. Use the ordinal covering of the ideal I_{K} to find a condition B_{2} ∈ P_{K} ∩ M with B_{2} ⊂ B_{1} and ∀s⃗ ∈ B_{2} g_{1}(s⃗) ⊂ g_{0}(s⃗).

Finally, let C ⊂ B be the set {r⃗ ∈ ℝ^{L} : r⃗ ↾ K ∈ B_{2} ∧ r⃗ ↾ L \ K ∈ g_{1}(r⃗ ↾ K)}. Clearly, this is a condition in the poset P_{L} ∩ M which forces that the filter obtained from the generic sequence r⃗_{gen} ↾ L \ K will meet the dense set Ḋ in the condition ġ_{1}(ṙ_{gen} ↾ K).

A genericity argument now concludes the proof of the claim.

Now work in the model V [G_{K}]. It is the case that X ∈ Mod, therefore by the definition of the set Mod and the previous claim the real r⃗_{gen}(i) is P_{I} ∩M-generic and it falls out of all I-small sets coded in the model V [G_{K}], in particular, it falls out of the set f(r⃗_{gen} ↾ K) as desired.

And finally it is possible to verify is that the partial order P_{L} really is an iteration of P_{I} along the ordering L.

Lemma 5.19. Let r⃗_{gen} be the P_{L}-name for the generic L-sequence of reals. For every index i ∈ L the real r⃗_{gen}(i) is P_{I}-generic over the model V [r⃗_{gen} ↾< i].

Proof. Fix an index i ∈ L. Suppose that D is an arbitrary condition in the poset P_{L} and Ȯ is a name such that D ⊩ Ȯ ⊂ Ṗ_{I} is an open dense set in the model V [r⃗_{gen} ↾< i]. By 5.9 Ȯ can be assumed to be a P_{L}-name. As in the previous proof, for every sequence s⃗ ∈ ℝ^{<i} define h(s⃗) = D(s⃗) ↾↾ {i} = {r ∈ ℝ : the set {t⃗ ∈ ℝ^{L\(K∪{i})} : s^{⌢}r^{⌢}t⃗ ∈ D} is I^{L\(K∪{i})}-positive}. The condition D ↾↾< i forces in the poset P_{<i} the set Ḋ(r⃗_{gen} ↾ i) to be I-positive, and so there is a name ṗ for a condition in the set Ȯ below it. By 2.2 there is a condition E^{∗} ⊂ D ↾↾< i and a Borel function f : E^{∗} → P_{I} such that E^{∗} forces ṗ = ḟ(r⃗_{gen} ↾< i). Let E = {r⃗ ∈ D : r⃗ ↾< i ∈ E^{∗} and r⃗(i) ∈ ḟ(r⃗ ↾< i)}. It is not difficult to verify using the previous lemma and 5.9 that this is an I^{L}-positive Borel set. Clearly E forces the real r⃗_{gen}(i) to meet the open dense set Ȯ.

The attentive reader must of course wonder how the notion of iteration we have just developed compares with the countable support iteration in the case that L is well-ordered. The final lemma in this subsection validates the expectation: they are the same.

Lemma 5.20. Let I be a σ-ideal σ-generated by a projective collection of closed sets, let L be a countable well-ordering and let α be its length; identify L and α. A Borel set B ⊂ ℝ^{L} is in the ideal I^{L} (as defined in this Subsection) if and only if it is in the Fubini power I^{α}.

Proof. Suppose first that the Borel set B is in the ideal I^{α}. Then Adam has a winning strategy σ in the game G(B), and by the Real Determinacy in the model L(ℝ)(Γ), such a strategy can be found in the model L(ℝ)(Γ). The strategy immediately breaks up into functions g_{i} : ℝ^{<i} → I, i ∈ L in the model such that for every sequence r⃗ ∈ B there is an index i ∈ L such that r⃗(i) ∈ g_{i}(r⃗ ↾< i). Thus B ∈ I_{0}^{L} ⊂ I^{L}.

Now suppose that the Borel set B is not in the ideal I^{α}. Consider the iteration P_{I^{α}} of the poset P_{I}. This is a proper notion of forcing, B is a condition in it and it forces the generic sequence to be in the set Ḃ. By Claim 5.12, B is I^{L}-positive as desired.

The argument makes it clear that I^{α} ∩ L(ℝ)(Γ) = I^{L}.

5.3 The uncountable length

Now suppose that L is an arbitrary linear ordering. An iteration P_{L} of the forcing P along the ordering L can be defined in the following way. A condition is a set C such that for some countable subset K ⊂ L called the domain of C the set C ⊂ ℝ^{K} is Borel and I^{K}-positive. The ordering is defined by C ≤ B if dom(B) ⊂ dom(C) and ∀r⃗ ∈ C r⃗ ↾ dom(B) ∈ B. The analysis performed in Subsection 4.1 must now be repeated to show that the forcing has the expected properties, in particular

Lemma 5.21. The P_{L}-extension is given by an L-sequence r⃗_{gen} of reals in such a way that a condition B is in the generic filter if and only if r⃗_{gen} ↾ dom(B) ∈ Ḃ.

Lemma 5.22. For every initial segment K ⊂ L the iteration P_{K} is a regular subordering of P_{L}.

Lemma 5.23. For every countable elementary submodel M of a large enough structure and every condition B ∈ M the set C ⊂ ℝ^{M∩L} of all M-generic sequences is an I^{M∩L}-positive Borel set. In particular,

1. the iteration is proper

2. for every condition B and every name ẋ for a real there is a stronger condition C ≤ B and a Borel function f : C → ℝ such that C ⊩ ẋ = ḟ(r⃗_{gen} ↾ dom(C)).

Lemma 5.24. For every index i ∈ L the real r⃗_{gen}(i) is forced to be P_{I}-generic over the model V [r⃗_{gen} ↾< i].

All the work is just a repetition of the arguments from the previous Subsection and Subsection 4.1 and as such is left to the reader. It is a rather good exercise to go through the proofs in detail. The important point is that the lemmas must be proved in the order indicated, in particular the “iteration” property comes the very last.

As in the case of the countable orderings, for uncountable wellorderings L the notions of an iteration of the poset P along the ordering L obtained in this subsection and in Section 4 coincide. Also, given a countable linear ordering K and an ordinal α, the countable support iteration of the poset P_{K} along the ordinal α is exactly equal to the iteration of the poset P along the ordering consisting of α many copies of K.

5.4 Sacks forcing iteration

In [9] Kanovei defined an iteration L − Sacks of Sacks forcing along an arbitrary countable linear order L. A condition in L − Sacks is a set C ⊂ ℝ^{L} which is the range of a homeomorphism π : ℝ^{L} → C such that π(r⃗) ↾ K depends only on r⃗ ↾ K for every sequence r⃗ ∈ ℝ^{L} and every initial segment K ⊂ L. Kanovei calls such homeomorphisms projection-keeping. The ordering on L − Sacks is that of inclusion.

Theorem 5.25. For every countable linear ordering L the posets L−Sacks and P_{ctble}^{L} are forcing equivalent.

In order to prove this I will argue that the range of a projection keeping homeomorphism is an ctble^{L}-positive Borel set, and conversely, that every ctble^{L}-positive Borel set has a subset which is a range of a projection-keeping homeomorphism. As a side benefit, this will show that the Borel part of the ideal ctble^{L} is a projectively defined ideal: its elements are those subsets of ℝ^{L} with a Borel superset that does not contain the range of any projection-keeping homeomorphism. For notational convenience, put ℝ = 2^{ω}.

So let C ⊂ ℝ^{L} be the range of a projection-keeping homeomorphism. It is not difficult to see that in the L − Sacks forcing C ⊩ r⃗_{gen} ∈ Ċ and therefore the set C must be ctble^{L}-positive. On the other hand, let C be an ctble^{L}- positive Borel set; we must find a projection-keeping homeomorphism whose range is included in the set C. First, a bit of notation. Let U_{i} : i ∈ L be distinct copies of the set 2^{<ω} and let σ : ω → ⋃_{i} U_{i} be an enumeration which respects the extension ordering on each U_{i}. For every integer n let L_{n} be the set of all i ∈ L such that rng(σ ↾ n) ∩ U_{i} ≠ 0, for each i ∈ L_{n} let S_{n}^{i} be the tree rng(σ ↾ n) ∩ U_{i} and let T_{n}^{i} be the set of its terminal nodes.

Fix a countable elementary submodel M of a large enough structure containing all the relevant information and enumerate all open dense subsets of the forcing P_{L} in the model M by O_{n} : n ∈ ω. By induction on n ∈ ω construct functions f_{n} and g_{n} in the model M, with domain ∏_{i∈L_{n}} T_{n}^{i} so that

1. f_{n} : ∏_{i∈L_{n}} T_{n}^{i} → ∏_{i∈L_{n}} U_{i} is a function such that

• f_{n}(t⃗)(i) depends only on t⃗ ↾ i for every t⃗ ∈ dom(f_{n}) and i ∈ L_{n}

• if t⃗_{0}, t⃗_{1} are distinct elements of ∏_{i∈L_{n}} T_{n}^{i} and i ∈ L_{n} is the smallest index at which they differ then f_{n}(t⃗_{0})(i) and f_{n}(t⃗_{1})(i) are incompatible elements of U_{i}–by the previous item i is the smallest index at which this can happen • if m ∈ n are integers and t⃗_{0} ∈ ∏_{i∈L_{m}} T_{m}^{i}, t⃗_{1} ∈ ∏_{i∈L_{n}} T_{n}^{i} are sequences such that for every index i ∈ L_{m} the sequence t⃗_{1}(i) extends t⃗_{0}(i) then f_{m}(t⃗_{0})(i) ⊂ f_{n}(t⃗_{1})(i).

2. g_{n} : ∏_{i∈L_{n}} T_{n}^{i} → P_{ctble}^{L} is a function such that

• the functional values of g_{n} are conditions in the open dense set ⋃_{m∈n} O_{m} below C. Moreover, for every sequence t⃗ ∈ dom(g_{n}) and every index i ∈ L_{n} = dom(t⃗), every sequence s⃗ ∈ g_{n}(t⃗) has f_{n}(t⃗)(i) ⊂ s⃗(i).

• g_{n} respects projections, that is, g_{n}(t⃗) ↾↾≤ i depends only on t⃗ ↾≤ i for every t⃗ ∈ dom(g_{n}) and i ∈ L_{n}

• if m ∈ n are integers and t⃗_{0} ∈ ∏_{i∈L_{m}} T_{m}^{i}, t⃗_{1} ∈ ∏_{i∈L_{n}} T_{n}^{i} are sequences such that for every index i ∈ L_{m} the sequence t⃗_{1}(i) extends t⃗_{0}(i) then f_{m}(t⃗_{0}) ≥ f_{n}(t⃗_{1}).

The functions f_{n} are finite approximations to the future homeomorphism while the g_{n}’s will ensure that the range of the homeomorphism is included in the set C. Suppose that the induction has been completed. For a sequence r⃗ ∈ ℝ^{L} let π(r⃗) be the unique sequence s⃗ ∈ ℝ^{L} such that for every integer n, if t⃗ ∈ ∏_{i∈L_{n}} T_{n}^{i} is the unique element such that for every i ∈ L_{n} t⃗(i) ⊂ r⃗(i), then f_{n}(r⃗)(i) ⊂ s⃗(i). Then π is a well-defined continuous function by the third item of (1), it is one-to-one by the second item and by the first item it is projectionkeeping. The range of π is included in the set C: for every sequence r⃗ ∈ ℝ^{L} the set {g_{n}(t⃗) : t⃗ ∈ ∏_{i∈L_{n}} T_{n}^{i}, ∀i ∈ L_{n} t⃗(i) ⊂ r⃗(i)} forms a decreasing chain by the third item of (2) and the first item says that this chain is M-generic, the generic real belongs to the set C by Lemma 2.1 applied in the model M and it is exactly π(r⃗). The second item of (2) is there just to keep the induction going.

Set f_{0} = g_{0} = 0. The functions f_{n+1} and g_{n+1} are obtained from f_{n} and g_{n} using a rather simple surgery and the following claim. The details are left to the reader.

Claim 5.26. Let i_{0} > i_{1} > ⋯ ≥ i_{m̄−1} be an enumeration of the elements of L_{n} in decreasing order and let O ⊂ P_{ctble}^{L} be an open dense set. For every integer m ≤ m̄, every sequence u⃗ ∈ ∏_{i∈L_{n},i≤i_{m}} T_{n}^{i} and every map h : {t⃗ ∈ ∏_{i∈L_{n}} T_{n}^{i} : u⃗ ⊂ t⃗} → P_{ctble}^{L} which respects projections, there is a projection respecting map k with the same domain such that k(t⃗) ≤ h(t⃗) is an element of the set O for every sequence t⃗ ∈ dom(h).

Proof. By induction on m. The case m = 0 is trivial–the function h will have only one sequence in its domain. Now suppose the claim has been verified for m. For m + 1, fix an arbitrary sequence u⃗ and a projection respecting map h as in the statement of the claim. Let u⃗_{0}, u⃗_{1}, … u⃗l̄ be an enumeration of all sequences in which extend the sequence u⃗. By induction on l ∈ l̄construct I^{≤i_{m+1}}-positive sets B_{l} and projection respecting maps k_{l} : {t⃗ ∈ ∏_{i∈L_{n}} T_{n}^{i} : u⃗_{l} ⊂ t⃗} → O such that • B_{0} = ℝ^{≤i_{m+1}}

• k_{l} is a function obtained from the induction hypothesis applied to the sequence u_{l} and the function h_{l} : {t⃗ ∈ ∏_{i∈L_{n}} T_{n}^{i} : u⃗_{l} ⊂ t⃗} → P defined by h_{l}(t⃗) = h(t⃗) ∩ {r⃗ ∈ ℝ^{L} : r⃗ ↾≤ i_{m+1}}.

• B_{l+1} is the common projection of all the conditions in the range of k_{l} to the initial segment ≤ i_{m+1}.

This is easy to do; note that the sets B_{l} for l > 0 are all subsets of the common projection of the conditions h(t⃗) to the initial segment ≤ i_{m+1} and so the second item makes sense. Let k : {t⃗ ∈ ∏_{i∈L_{n}} T_{n}^{i} : u⃗ ⊂ t⃗} → P_{ctble}^{L} be the map defined k(t⃗) = k_{l}(t⃗) ∩ {r⃗ ∈ ℝ^{L} : r⃗ ↾≤ i_{m+1} ∈ Bl̄} where l ∈ l̄ is the unique number such that u⃗_{l} ⊂ t⃗. It is easy to verify that k is the desired function.

This concludes the proof of Theorem 5.25. Before I leave the subject I will pause to investigate the forcing properties of the posets P_{L} and compare the cases of wellfounded and illfounded ordering L.

Corollary 5.27. The ideal ctble^{L} is homogeneous for every countable linear ordering L.

Proof. If B ⊂ ℝ^{L} is a Borel positive set, then any projection-keeping homeomorphism h : ℝ^{L} → rng(h) ⊂ B has the property that the preimages of ctble^{L}-small sets are ctble^{L}-small. To see this, define the ideal J on rng(h) by A ∈ J if h^{−1}A ∈ ctble^{L}. It is enough to show that ctble^{L} ⊂ J. Now ctble_{0}^{L} ⊂ J follows from the projection keeping property of the function h, and it is clear that the ideal J is ordinal covering in the model L(ℝ)(Γ). Thus ctble^{L} ⊂ J and the corollary follows.

The proof of Theorem 5.25 also shows that the poset P_{ctble}^{L} has the Sacks property [31] and therefore has very strong preservation properties. (Aside: It is no coincidence that a proof of homogeneity would have similar consequences.) If L is actually a wellordering, the argument in Section 7.1 gives almost the strongest preservation theorem possible. If L is illfounded, the best theorem available is

Theorem 5.28. Suppose that 𝔵 is a tame invariant such that ZFC+LC proves that there is a c.c.c. poset Q ∈ L(ℝ) forcing 𝔵 = ℵ_{1}. Then ZFC+LC+CH proves that for every linear ordering L the iteration of Sacks forcing along L forces 𝔵 = ℵ_{1}.

Proof. Working in the context of ZFC+LC+CH it will be enough to find a proper forcing extension with 𝔵 < cov(ctble^{ℚ}). The rest of the proof follows the lines of the argument for Theorem 7.7 using the fact that every countable linear ordering embeds into the rationals. To get the desired extension, first find a proper forcing extension V [G] in which cov(ctble^{ℚ}) = ℵ_{2}–for example iterate the ℚ-iteration of Sacks forcing ℵ_{2} many times. Now let V [G][H] be the extension using the c.c.c. definable forcing Q which makes 𝔵 = ℵ_{1}. It will be enough to show that cov(ctble^{ℚ}) = ℵ_{2} in V [G][H]. Suppose {A_{γ} : γ ∈ ω_{1}} are ctble^{ℚ}-small sets in the class Γ. Similarly to Claim 4.25 it is possible to show that each of the sets A_{γ} is a union of ℵ_{1} = 𝔡𝔢𝔱 many sets of the form A_{g}; thus we may assume that for every ordinal γ ∈ ω_{1} there is a function g_{γ} ∈ Γ such that for some rational number i g_{γ} : ℝ^{<i} → ℝ and A_{γ} = {r⃗ ∈ ℝ^{ℚ} : r⃗(i) = g_{γ}(r⃗ ↾< i)}. Now V [G][H] is a definable c.c.c. extension of the model V [G], therefore in the model (L(ℝ)(Γ))^{V [G]} there are functions h_{γ} : γ ∈ ω_{1} such that for every ordinal γ ∈ ω_{1} and every rational number i and every sequence r⃗ ∈ ℝ^{<i} the value h_{γ}(r⃗) is a countable set of reals and if r⃗ ∈ dom(g_{γ}) and g_{γ}(r⃗) ∈ V [G] then g_{γ}(r⃗) ∈ h_{γ}(r⃗). Clearly, the collection of sets {B_{γ} = {r⃗ ∈ ℝ^{ℚ} : ∃i r⃗(i) ∈ h_{γ}(r⃗ ↾< i} : γ ∈ ω_{1}} consists of ctble^{ℚ}-small sets and the whole collection is in the model V [G]. Thus there must be a sequence r⃗ ∈ ℝ^{ℚ} ∩ V [G] avoiding all of these ℵ_{1} many sets, and it will avoid all the sets {A_{γ} : γ ∈ ω_{1}} as well.

The class of invariants satisfying the assumptions of the theorem includes 𝔞, 𝔟, 𝔡 and 𝔱 among others, as the reader can easily check. However, it is not as large as the class of all tame invariants which are < 𝔠 in some extension, and that makes the theorem weaker than the corresponding case of Theorem 7.4 for a wellfounded iteration. At the root of the problem lies the following argument. It provides a simple covering number 𝔵 such that cov(ctble^{L}) ≤ 𝔵 ≤ 𝔠 = cov(ctble^{K}) for all countable linear orderings L, K, L illfounded and K wellfounded, where the second inequality cannot be reversed. It also shows that the assumption in 5.28 that Q ∈ L(ℝ) cannot be dropped.

Example 5.29. Define J be the σ-ideal on ℝ^{ω} σ-generated by the sets A_{g} = {r⃗ ∈ ℝ^{ω} : ∃i ∈ ω r⃗(i) = g(r⃗ \ i + 1)} as g varies over all Borel functions from ℝ^{ω} to ℝ. Then for every illfounded countable linear order L cov(ctble^{L}) ≤ cov(J) and in ZFC there is a c.c.c. forcing making cov(J) = ℵ_{1}.

Proof. Clearly, every countable illfounded linear order L contains a copy of ω^{∗} (ω with a reverse ordering) and cov(ctble^{L}) ≤ cov(ctble^{ω^{∗}}) ≤ cov(J) essentially by the definitions. Thus I only need to produce a c.c.c. forcing Q making cov(J) = ℵ_{1}. The poset Q is a finite support iteration of length ω_{1} obtained in the following way. Suppose the iteration up to α ∈ ω_{1} has been found and V [G_{α}] is the corresponding extension. Inside this model use Example 5.15 to find a (non-Borel) function f_{α} from ℝ^{ω} → ℝ such that for every sequence r⃗ ∈ ℝ^{ω} there is an integer i such that r⃗(i) = f_{α}(r⃗ \ i + 1). The α-th iterand in the forcing Q to be some c.c.c. forcing in V [G_{α}] introducing a Borel function g_{α} such that f_{α} ⊂ g_{α}–such forcing is found in Claim B.13. It is not difficult to see that Q forces ℝ^{ω} to be covered by the ℵ_{1} many sets A_{g_{α}}, α ∈ ω_{1}.

The covering numbers of the ideals ctble^{L} for various countable linear orders can be compared using the following fact stated without proof.

Fact 5.30. For two countable linear orders L and K the following are equivalent:

• L can be embedded into K • there is a Borel function f : ℝ^{K} → ℝ^{L} such that the preimages of ctble^{L}-

small sets are ctble^{K}-small.

Moreover, L can be embedded into any countable wellordered stack of copies of K if and only if ZFC+LC proves cov(ctble^{L}) ≥ cov(ctble^{K}).

6 Towers of ideals

In certain applications, it is desirable to produce partial orders adding objects more complicated than reals–for example subsets of ω_{1}. In such cases it is necessary to generalize the machinery presented in the previous sections.

Definition 6.1. For every countable ordinal α the symbol 𝔽_{α} denotes the collection of all continuous increasing functions from α into itself, viewed as a Borel subset of the space P(α × α).

Definition 6.2. For every countable ordinal α let I_{α} be a σ-ideal on the set 𝔽_{α}. The collection {I_{α} : α ∈ ω_{1}} is called a tower if

• for all countable ordinals β ∈ α and all Borel I_{α}-positive sets C ⊂ 𝔽_{α} such that every function in C sends β to β, the set C ↾ β = {f ↾ β : f ∈ C} is I_{β}-positive.

• for all countable ordinals β ∈ α and all Borel I_{β}-positive sets B ⊂ 𝔽_{β}, the set {f ∈ 𝔽_{α} : f ↾ β ∈ B} is I_{α}-positive.

Definition 6.3. Given a tower T = {I_{α} : α ∈ ω_{1}} of ideals the partial order P_{T} consists of all sets B such that for some ordinal α ∈ ω_{1}, B is a Borel I_{α}- positive subset of 𝔽_{α}; I will write α = α(B). The ordering is defined by C ≤ B if α(B) ≤ α(C) and for every function f ∈ C, f ↾ α(B) ∈ B.

Definition 6.4. Given a tower T of ideals, the P_{T} -name Ḟ_{gen} for the generic function from ω_{1} to itself is defined by ⟨β̌, γ̌⟩ ∈ Ḟ_{gen} if there is some condition B in the generic filter such that β, γ ∈ α(B) and ⟨β, γ⟩ ∈ f for every function f ∈ B.

There are several obvious degrees of freedom in the above definitions–the ideals I_{α} can measure perhaps subsets of P(α × α) or subsets of ω^{α} etc. I chose the case of continuous increasing functions from α to α only because it seemed to simplify the notation somewhat in the applications. The theory transfers without changes to the other cases. Another possible minor change could result if the ideals are not indexed by all countable ordinals but by ordinals coming from some closed unbounded set.

It is not difficult to verify that Ḟ_{gen} is indeed a name for a function from ω̌_{1} into itself. The basic fact is that the generic extension for a tower poset is determined by this function.

Lemma 6.5. Let T be a tower of ideals. The partial ordering P_{T} forces the following. For every condition C ∈ P̌_{T}, C is in the generic filter if and only if Ḟ_{gen} ↾ α(C) ∈ C.

Proof. For the left-to-right direction argue that any condition B forces Ḟ_{gen} ↾ α(B) ∈ Ḃ. For this is enough to show that for every countable ordinal β, every Borel subset B ⊂ P(β ×β) and every condition C ∈ P_{T} such that β ≤ α(C) and C ↾ β ⊂ B, the condition C forces Ḟ_{gen} ↾ β ∈ Ḃ. This is proved by induction on the Borel complexity of the set B. It is quite clear for closed and basic open sets B. If B = ⋂_{n} B_{n} and C ∈ P_{T} is a condition such that C ↾ β ⊂ B, then by the induction hypothesis C forces Ḟ_{gen} ↾ β̌ ∈ ⋃_{n} Ḃ_{n} = Ḃ. And if B = ⋃_{n} B_{n}, C ∈ P_{T} is a condition such that C ↾ β ⊂ B and D ≤ C is an arbitrary condition with α(D) = α, then one of the Borel sets D_{n} = {f ∈ D : f ↾ β ∈ B_{n}} must be I_{α}-positive and by the induction hypothesis it forces Ḟ_{gen} ↾ β ∈ Ḃ_{n}. Thus C ⊩ Ḟ_{gen} ↾ β̌ ∈ Ḃ as desired.

For the right-to-left direction first argue that if β is a countable ordinal and B ⊂ 𝔽_{β} is a I_{β}-small set then P_{T} ⊩ Ḟ_{gen} ↾ β ∉ Ḃ. For suppose that C ∈ P_{T} is an arbitrary condition, and strengthening if necessary assume that β ∈ α = α(C). There are two cases. Either the Borel set C_{0} = {f ∈ C : β is not closed under the function f} is I_{α}-positive, and by the previous paragraph it then is a condition stronger than C which forces Ḟ_{gen} ∉ Ḃ. If this fails, then the set C_{1} = {f ∈ C : β is closed under f} must be positive. The Borel set D = {f ∈ C_{1} : f ↾ β ∈ B} is I_{α}-small, since the ideals form a tower. Thus again, C_{1} \ D is a condition stronger than C which by the previous paragraph forces Ḟ_{gen} ↾ β ∉ Ḃ as desired.

The right-to-left direction now immediately follows. Every condition C ∈ P_{T} forces Ḟ_{gen} ↾ dom(C) ∈ Ċ by the first paragraph. On the other hand, if Ḟ_{gen} ↾ α(C) ∈ Ċ as forced by some condition D with α(C) ∈ α = α(D), then the Borel set E = {f ∈ D : f ↾ α(C) ∈ Ċ} must be I_{α}-positive by the second paragraph, since the condition D forces Ḟ_{gen} ↾ α̌ ∈ Ė. Then E is a strengthening of the condition D which is also stronger than C and so forces C into the generic filter.

It is now quite easy to see that P_{T} forces Ḟ_{gen} to be a continuous increasing function from ω̌_{1} to ω̌_{1}.

Corollary 6.6. Let T be a tower of ideals. The partial ordering P_{T} is proper if and only if for every countable elementary submodel M of a large enough structure and every condition B ∈ P_{T} ∩ M the set C = {f ∈ 𝔽_{M∩ω_{1}} : f ↾ dom(B) ∈ B and f is M-generic for the poset P_{T} } is I_{M∩ω_{1}}-positive.

Proof. For the sake of Brevity write α = M ∩ ω_{1}. The set C ⊂ 𝔽_{α} is Borel by an argument parallel to Lemma 2.2. If C is I_{α}-positive, then C ∈ P_{T} is a condition stronger than B and by the previous lemma it forces Ḟ_{gen} ↾ α to be M-generic, in other words it is an M-master condition below B. On the other hand, if there is any M-master condition D ≤ B then it forces Ḟ_{gen} ↾ α ∈ Ċ, which by the second paragraph of the previous proof means that C must be I_{α}-positive.

The countable support iteration of length κ of the partial order P_{T} is handled exactly as in Section 4.1. To facilitate the notation, whenever X ⊂ Y are sets of ordinals, β ≤ α are ordinals and f⃗ is a function with domain Y whose functional values are in turn functions with domain α, write f⃗ ↾ (X × β) for the function g⃗ with domain X such that for all ordinals δ ∈ X, g⃗(δ) = f⃗(δ) ↾ β. Let P_{κ}(T) be the partial ordering consisting of all Borel I_{α}-perfect sets C ⊂ (𝔽_{α})^{X} for all countable ordinals α and all countable sets X ⊂ κ. Write X = dom(C) and α = α(C). The ordering is defined by C ≤ B if dom(B) ⊂ dom(C), α(B) ≤ α(C) and for every sequence f⃗ ∈ C the sequence f⃗ ↾ (dom(B) × α(B)) is in the set B. The generic object is a function F⃗_{gen} with domain κ̌ whose functional values are functions from ω̌_{1} to ω̌_{1} defined by F⃗_{gen}(δ̌)(β̌) = γ̌ if there is a condition B in the generic filter with δ ∈ dom(B), β, γ ∈ α(B) and for all sequences f⃗ ∈ B, f⃗(δ)(β) = γ.

Lemma 6.7. (ZFC+LC) Suppose that T = {I_{α} : α ∈ ω_{1}} is a projectively defined tower of ideals such that the posets P_{T} as well as P_{I_{α}} for all α ∈ ω_{1} are provably proper. Then:

1. P_{κ}(T) is forcing equivalent to the countable support iteration of P_{T} of length κ and F⃗_{gen} is the sequence of the generic functions.

2. P_{κ}(T) forces that a condition B is in the generic filter if and only if F⃗_{gen} ↾ (dom(B) × α(B)) ∈ Ḃ.

3. For every condition B ∈ P_{κ}(T) and every name ẋ for a real there is a strengthening C ≤ B and a Borel function f : C → ℝ such that C ⊩ ẋ = ḟ(F⃗_{gen} ↾ (dom(C) × α(C))).

The proof is a verbatim repetition of arguments from Subsection 4.1. The properness of the posets P_{I_{α}} is used to argue that for all countable ordinals α and γ, the Borel subsets of (𝔽_{α})^{γ} with no I_{α}-perfect subset form a σ-ideal basis.

Though nothing can compare to the joys of unillustrated pure theory, I will now spoil the fun and describe two examples relevant in the applications.

6.1 Shooting a club with no infinite subset in the ground model

Let α be a countable indecomposable ordinal. Consider the σ-ideal I_{α} on 𝔽_{α}: a set is in the ideal I_{α} if it has a Borel superset with no perfect subset consisting of functions with mutually almost disjoint ranges. Here almost disjoint means “with finite intersection”.

Lemma 6.8. I_{α} is a proper σ-ideal and the ideals I_{α} form a tower.

Proof. To see that I_{α} is a σ-ideal, suppose that B_{n} : n ∈ ω are Borel sets, each without a perfect subset of functions with mutually almost disjoint ranges. Then their union does not contain such a set either. If it did contain such a subset C then for some integer n the set C ∩ B_{n} would be uncountable and therefore it would contain a perfect subset, contradicting the assumption on the set B_{n}.

To see that the ideals I_{α} form a tower, let β ∈ α be countable indecomposable ordinals and C ⊂ 𝔽_{α} be an I_{α}-positive set such that all functions in it map β to β. Let D ⊂ C be a perfect subset consisting of functions with almost disjoint ranges. The set {f ↾ β : f ∈ D} is an uncountable analytic set consisting of functions with almost disjoint ranges, and as such contains a perfect subset. Thus the set is I_{β}-positive as desired.

It is not difficult to see that every Borel I_{α}-positive set contains some perfect subset C consisting of functions with mutually almost disjoint ranges. Then also C is I_{α}-positive together with all of its Borel uncountable subsets. Thus densely the posets P_{I_{α}} are forcing equivalent to Sacks forcing and

Lemma 6.9. For every indecomposable countable ordinal α the forcing P_{I_{α}} is proper.

Of course the interest here lies not in the posets P_{I_{α}} but in the forcing P_{T} where T is the tower of the ideals I_{α}.

Lemma 6.10. The forcing P_{T} is proper.

Proof. Let ⟨M_{n} : n ∈ ω⟩ be an ∈-tower of countable elementary submodels of a large enough structure, with M = ⋃_{n} M_{n}, and let B ∈ P_{T} ∩ M_{0}. I need to find an M-master condition below B. By Corollary 6.6 it is enough to produce a perfect set of M-generic continuous increasing functions from M ∩ ω_{1} into itself with mutually almost disjoint ranges.

To do this, fix an injection e : 2^{<ω} → ω such that e(s) is greater than the length of s and t ⊂ s implies e(t) ∈ e(s) for all sequences t and s. Fix also a listing {D_{n} : n ∈ ω} of all open dense subsets of the poset P_{T} in the model M such that D_{n} ∈ M_{n} for all integers n. To simplify the notation, let δ_{n} = M_{n} ∩ ω_{1} and δ = M ∩ ω_{1}. By induction on the length of a finite sequence s ∈ 2^{<ω} construct conditions B_{s} ∈ P_{T} ∩ M so that:

• B = B_{0} and for sequences t ⊂ s, B_{t} ≥ B_{s}. Moreover, if the length of s is n + 1 then B_{s} ∈ D_{n}. The intention here is that for every real r ∈ 2^{ω} the sequence {B_{s} : s ⊂ r} will be M-generic, giving a continuous increasing function f_{r} from δ into itself. The functions {f_{r} : r ∈ 2^{ω}} will form my perfect set of generics with mutually almost disjoint ranges.

• For every nonzero sequence s the ordinal α_{s} = dom(B_{s}) is between δ_{e(s)} and δ_{e(s)+1}. Moreover, if 0 = t_{0} ⊂ t_{1} ⊂ t_{2} ⊂ ⋯ ⊂ t_{n} = s lists all the initial segments of s, then the range of every function in B_{s} is a subset of [0, α_{t_{0}}] ∪ [δ_{e(t_{1})}, α_{t_{1}}] ∪ [δ_{e(t_{2})}, α_{t_{2}}] ∪ ⋯ ∪ [δ_{e(t_{n})}, α_{t_{n}}]. This condition will imply that if r_{0} ≠ r_{1} are two distinct reals, then writing s for their longest common initial segment, rng(f_{r_{0}}) ∩ rng(f_{r_{1}}) ⊂ α_{s} + 1.

• For every sequence s and all functions g ∈ B_{s^{⌢}0} and h ∈ B_{s^{⌢}1}, the functions g ↾ α_{s} and h ↾ α_{s} have almost disjoint ranges. This condition will imply that if r_{0} ≠ r_{1} are two distinct reals, then writing s for their longest initial segment it is the case that the functions f_{r_{0}} ↾ α_{s} and f_{r_{1}} ↾ α_{s} have almost disjoint ranges. Together with the previous item this will mean that the functions f_{r_{0}} and f_{r_{1}} themselves have almost disjoint ranges, completing the proof of the lemma.

To perform the inductive step, suppose that s is a sequence and the condition B_{s} has been obtained. To get the conditions B_{s^{⌢}0} and B_{s^{⌢}1}, first find in B_{s} a perfect subset consisting of functions with mutually almost disjoint ranges and divide it into two uncountable Borel pieces C_{0} and C_{1}. It will be the case that C_{0} ≥ B_{s^{⌢}0} and C_{1} ≥ B_{s^{⌢}1}, securing the third item above. To get the condition B_{s^{⌢}0}, consider the ordinal δ_{e(s^{⌢}0)} · ω and the set C̄_{0} of continuous increasing functions g of it into itself such that g ↾ α_{s} ∈ C_{0}, g(α_{s}) = α_{s} and g(α_{s} + 1) = δ_{e(s^{⌢}0)}. It is not difficult to verify that C̄_{0} is indeed a condition in the poset P_{T} and C̄_{0} ∈ M_{e(s^{⌢}0)+1}. Just let B_{s^{⌢}0} be any condition in this model stronger than C̄_{0} in the dense set D_{lth(s)}. The condition B_{s^{⌢}1} is obtained similarly.

The basic effect of the forcing just introduced is stated in the obvious

Lemma 6.11. The P_{T} generic object is a continuous increasing function from ω_{1} into itself. The range of the function does not contain any ground model infinite subset.

Proof. Only the second sentence needs a proof. Suppose that x ⊂ ω_{1} is infinite and B ∈ P_{T} is a condition. Strengthening the condition if necessary we may assume that there are infinitely many elements of x below dom(B) and B consists of functions with mutually almost disjoint ranges. This means that there is at most one function f ∈ B whose range contains x ∩ dom(B) as a subset. Then C = B \ {f} is a condition strengthening B which consists only of functions whose ranges do not include x ∩ dom(B) as a subset and by Lemma 6.5 it forces that the range of the generic function does not include x as a subset.

And the behavior of the iteration is governed by the following

Lemma 6.12. Let α be a countable indecomposable ordinal and γ be a countable ordinal. For every collection of < 𝔠-many projective subsets of the space (𝔽_{α})^{γ}, each of them without a Borel I_{α}-perfect subset, even its union does not contain such a subset.

Proof. The idea behind the lemma is the following. For every Borel I_{α}-positive set Y ⊂ 𝔽_{α} there is a Borel I_{α}-positive set Z ⊂ Y consisting of functions with mutually almost disjoint ranges. Then by the perfect set theorem, every universally Baire uncountable subset of Z has a perfect subset, which then constitutes a Borel I_{α}-positive subset of the universally Baire set. In other words, densely in the forcing P_{I_{α}} the universally Baire part of the ideal I_{α} consists of just the countable sets and we are in the case of Subsection 4.3. To translate this idea into a proof of the claim, let B be a Borel I_{α}-perfect subset of (𝔽_{α})^{γ}. It is not difficult to thin out this set into a Borel I_{α} perfect set B′ such that for each ordinal β ∈ γ and each sequence f⃗ ∈ B′ ↾ β the set {g ∈ 𝔽^{α} : f⃗^{⌢}g ∈ B′ ↾ β + 1} consists of functions with mutually almost disjoint ranges. The ideal I^{γ}(P_{I_{α}}) below this set reduces to a Fubini power of the countable ideal on the space 𝔽_{α}, and therefore the set B′ cannot be covered by < 𝔠 many small projective sets by Corollary 4.16.

6.2 Shooting a club with finite intersection with every ground model ordertype ω set

Let α be a countable ordinal closed under ordinal exponentiation, greater than ω. Define an ideal J_{α} on 𝔽_{α}: a set B is in J_{α} if there are countably many subsets of α of ordertype ω such that the range of every function in the set B has infinite intersection with one of them.

Lemma 6.13. J_{α} is a proper σ-ideal and the ideals J_{α} form a tower.

Proof. A little bit of effort is needed to show that the ideal I_{α} is proper for all countable ordinals closed under ordinal exponentiation, greater than ω. The proof is left to the reader.

Let S be the tower of the ideals J_{α}. The treatment of the forcing P_{S} is very similar to the previous subsection.

Lemma 6.14. The posets P_{J_{α}} are proper.

The proof is just a simplification of the argument for

Lemma 6.15. The poset P_{S} is proper.

Proof. I will present an argument which is parallel to Lemma 3.6 and provides essentially no insight into the combinatorics of the poset. Consider the following game between Adam and Eve. Adam first plays a condition B_{0} ∈ P_{S} and then on a fixed schedule produces (one by one) open dense subsets O_{n} ⊂ P_{s}, (one by one) limit ordinals β_{n} and (point by point) cofinal subsets x_{n} ⊂ β_{n} of ordertype ω, for n ∈ ω. Eve produces on her own pace a descending chain of conditions B_{0} ≥ B_{1} ≥ B_{2} ≥ …, meaning that she can wait for an arbitrary finite number of rounds before producing the next condition. Moreover, every time Eve plays a condition B_{n}, in the next round Adam indicates a bounded interval I_{n} of countable ordinals larger than α(B_{n}). Eve wins if, writing α_{n} for α(B_{n}) and α for the supremum of α_{n}’s, it is the case that the chain she produced meets all the open dense subsets Adam indicated, there is a unique function f : α → α such that for all numbers n f ↾ α_{n} ∈ B_{n} and the range of this function f has finite intersection with every set x_{n} Adam produced and empty intersection with all the intervals I_{n} Adam indicated.

The concept of a fixed schedule for Adam in this game needs elaboration. The game has as a parameter a ladder L, that is, a function which assigns every countable limit ordinal β a cofinal ω-sequence L(β) in it. At round n, Adam will play ordinal β_{n} and finite sets x_{m}^{n} for all m ≤ n such that x_{m}^{n} ⊂ L(β_{m})(n) and such that x_{m}^{k} = x_{m}^{l} ∩ L(β_{m})(k) whenever m ≤ k ≤ l. Of course the intention is that in the end x_{n} = ⋃_{k} x_{n}^{k} will be the desired cofinal subset of β_{n} of ordertype ω.

The key claim:

Claim 6.16. Eve has a winning strategy.

Proof. The basic observation is that the payoff set of the game is Borel in the (very large) space of all possible plays and therefore the game is determined [12] 20.5. Thus it is enough to derive a contradiction from the assumption that Adam has a winning strategy τ. The argument is parallel to the proof of the properness of Laver forcing. First observe that for every condition B ∈ P_{S} and every finite set {β_{m} : m ∈ n} of limit ordinals below α(B) there is an integer m = m(B, β_{0}, β_{1}, … β_{n−1}) such that for every collection of sets {x_{k} : k ∈ n}, each of them a cofinal subset of the interval [L(β_{k})(m), β_{k}) of ordertype ω, the Borel set {f ∈ B : ∀k ≤ n rng(f) ∩ x_{k} = 0} is still J_{α(B)}-positive. For if this failed for some B and finite collection of ordinals, it would be possible to find counterexamples {x_{k}^{m} : k ∈ n} for all integers m ∈ ω. Then the sets x_{k} = ⋃_{m} x_{k}^{m} all have ordertype ω and by the definition of the ideal J_{α(B)} there would be a positive set C ⊂ B and an integer m ∈ ω such that ∀f ∈ C∀k ∈ n rng(f)∩x_{k} ⊂ L(β_{k})(m), contradicting the choice of the sets x_{k}^{m} : k ∈ n.

Now I will produce a play against the strategy τ which results in Eve’s victory, proving the Claim. During the course of the play, Adam will produce a condition B_{0}, intervals I_{n}, ordinals β_{n}, sets x_{n}, and open dense sets O_{n}. Eve will produce a descending chain of conditions B_{n} ∈ P_{S} together with finite sets z_{n} of ordinals so that, writing k_{n} for the number of rounds played before Eve made the move B_{n}, the following holds for all n:

• m_{n} = m(B_{n}, β_{k} : k ∈ n) ∈ k_{n}

• ∀m ∈ n β_{m} ∈ α(B_{n}) and ∀f ∈ B_{n} rng(f) ∩ x_{m}^{k_{n}} ⊂ z_{m}

• ∀m ∈ n ∀f ∈ B_{n} rng(f) ∩ I_{m} = 0

• B_{n+1} ∈ O_{n}

If Eve succeeds in doing this, she wins: writing f for the function given by her descending chain, ∀m ∈ ω rng(f) ∩ x_{m} ⊂ z_{m} by the second item and ∀m ∈ ω rng(f)∩I_{m} = 0 by the third item as desired. The inductive construction of the play of course uses the first item to keep going. Suppose the condition B_{n} as well as all the sets z_{m} : m ∈ n and the play up to k_{n} have been constructed. Now look at the play that would result if from this point on Eve did not move and Adam kept on playing according to the strategy τ. Of course, Eve loses in this way, but more importantly Adam creates the interval I_{n} and some ordinals β̄_{m}, sets x̄_{m} ⊂ β̄_{m} and open dense sets Ō_{n}. There is a positive set C ⊂ B_{n} such that ∀k ∈ n ∀f ∈ C rng(f) ∩ x̄_{k} ⊂ L(β_{k})(m_{n}) by the choice of the integer m_{n}. There is a condition D ≤ C such that β_{n} ∈ α(D), I_{n} ⊂ α(D) and all functions in the set D have range disjoint from the interval I_{n}, and finally, there is a condition E ≤ D in the open dense set Ō_{n}. Now Eve will let her next move be B_{n+1} = E but it is important when she will play it: let m = m(E, β_{k} : k ∈ n + 1), let z_{n} = x̄_{n} ∩ L(β̄_{n})(m) and play the move B_{n+1} only after the strategy τ commits to the set Ō_{n}, the ordinal β̄_{n} and the sets x̄_{k} ∩ L(β̄_{k})(m) for all k ∈ n + 1. The induction hypotheses continue to hold.

The lemma follows from the claim. Fix Eve’s winning strategy σ. Let ⟨M_{i} : i ∈ ω^{2}⟩ be an arbitrary ∈-tower of countable elementary submodels of a sufficiently large structure containing σ. For the sake of Brevity write M for their union, δ_{i} = M_{i} ∩ ω_{1} and δ = M ∩ ω_{1}. To prove the properness it will be enough to find an M-master condition below an arbitrary condition B ∈ P_{S} ∩M. By Corollary 6.6 it is enough to show that the set C = {f ∈ 𝔽_{δ} : f ↾ α(B) ∈ B and f is M-generic} is J_{δ}-positive. So for every countable collection X of subsets of α of ordertype ω I must find a function f ∈ C whose range has finite intersection with all the sets in the collection. Divide the collection X into X_{0} and X_{1}, where X_{0} containing all the sets in X with supremum δ and X_{1} containing the rest of X_{0}. Since the sets in X_{0} have ordertype ω, it is possible to find an ω-sequence i_{0} ∈ i_{1} ∈ … of ordinals converging to ω^{2} such that every set in X_{0} is except for finitely many elements included in the union ⋃_{n} I_{n} where I_{n} is the interval [δ_{i_{n}+1}, δ_{i_{n+1}}). Now by careful bookkeeping it is possible to construct a play of the game such that

• Eve follows the strategy σ, Adam makes the first move B_{0} = B.

• writing p_{n} for the fraction of the play right before Eve plays the condition B_{n} we have that p_{n} ∈ M_{i_{n+1}}. After Eve plays B_{n} Adam will play I_{n} which by the previous sentence and the elementarity of the model M_{i_{n+1}} will be a legal move.

• in the course of the game Adam will produce all the sets in the collection X_{1} and all the open dense subsets of the poset P_{S} in the model M.

Write f : δ → δ for the function obtained from the play. The function f is M-generic by the third item and f ↾ α(B) ∈ B by the first item; thus f ∈ C. Moreover, for every set x ∈ X the intersection x ∩ rng(f) is finite. This follows from the third item for x ∈ X_{1} and from the second item and the choice of the ordinals i_{n} for x ∈ X_{0}. This completes the proof of the lemma.

Lemma 6.17. The generic object for the forcing P_{S} is a continuous increasing function from ω_{1} into itself. Its range has finite intersection with every ground model set of ordertype ω.

The cardinal characteristics governing the ideals J_{α} as well as the countable support iteration of the tower S are described by the following.

Lemma 6.18. Let α be a countable ordinal closed under ordinal exponentiation, larger than ω. Then cov(J_{α}) = add(J_{α}) = 𝔟.

Proof. I will need several definitions due to Todorcevic. Let L be a ladder on ω_{1}, that is, for every limit ordinal β ∈ ω_{1} L(β) is an increasing ω-sequence of ordinals converging to β. For countable ordinals δ ∈ γ ∈ ω_{1} the walk from γ to δ is a finite sequence γ = γ_{0} > γ_{1} > γ_{2} > ⋯ > γ_{n} = δ defined recursively so that γ_{m+1} is the predecessor of the ordinal γ_{m} if the latter is successor, and γ_{m+1} is the smallest element of the sequence L(γ_{m}) above the ordinal δ otherwise. The weight of the walk is the maximal number k such that for some m ∈ n, γ_{m+1} is k-th element of the sequence L(γ_{m}).

Now for every function g ∈ ω^{ω} and every countable limit ordinal β define the set x(g, β) ⊂ β by x(g, β)∩[L(β)(n), L(β)(n+1)) to be the set of those ordinals δ in the interval [L(β)(n), L(β)(n + 1)) such that the weight of the walk from L(β)(n+1) to δ is less than g(n). The set of walks from a fixed ordinal of weight less than a fixed number naturally forms a finitely branching tree without an infinite branch, therefore a finite tree. It follows that for every number n the set x(g, β) ∩ [L(β)(n), L(β)(n + 1)) is finite and so the set x(g, β) has ordertype ω. It is not difficult to see that if g < h for all but finitely many values, then x(g, β) ⊂ x(h, β) up to finitely many points, and for every cofinal set x ⊂ β of ordertype ω there is a function g such that x ⊂ x(g, β).

The lemma quickly follows. If {x_{γ} : γ ∈ κ} are < 𝔟 many subsets of a countable ordinal α with the corresponding suprema β_{γ}, for each of them choose a function g_{γ} ∈ ω^{ω} such that x_{γ} ⊂ x(g_{γ}, β_{γ}) and find a function h ∈ ω^{ω} modulo finite dominating all the functions {g_{γ} : γ ∈ κ}. Then the J_{α}-small sets B_{γ} = {f ∈ 𝔽_{α} : x_{γ} ∩ rng(f) is infinite} for γ ∈ κ are all subsets of the J_{α}-small set B = {f ∈ 𝔽_{α} : x(h, β) ∩ rng(f) is infinite for some limit ordinal β ≤ α}. This proves that add(J_{α}) ≥ 𝔟. To show that cov(J_{α}) ≤ 𝔟, choose an unbounded collection {g_{γ} : γ ∈ 𝔟} of functions in ω^{ω} and observe that the J_{α}-small sets B_{γ} = {f ∈ 𝔽_{α} : x(g_{γ}, α) ∩ rng(f) is infinite} cover the whole space 𝔽_{α}.

Corollary 6.19. Let α > ω be a countable ordinal closed under ordinal exponentiation, and let γ be an arbitrary countable ordinal. For every collection of < 𝔟-many projective subsets of the space (𝔽_{α})^{γ}, each without a Borel J_{α}-perfect subset, even its union does not contain such a subset.

Proof. In view of Corollary 4.16 or its proof, it is enough to show that every J_{α}-positive universally Baire set has a J_{α}-positive Borel subset. First argue by Lemma 6.18 and Lemma 2.12 (2) that the ideal J_{α} is suitably ordinally covering in every model of Real Determinacy and Dependent Choice, since every proper forcing adding a dominating real also adds a J_{α}-small set covering all ground model J_{α}-small sets. Then use Lemma 2.13 (2) and Fact C.8 to get the desired conclusion.

7 Applications

7.1 Isolating cardinal invariants

Traditionally, many cardinal invariants have certain forcing notions associated with them [31]. The following definition is an attempt to formalize this correspondence:

Definition 7.1. A cardinal invariant 𝔶 can be isolated if there is a partial order P_{𝔶} such that for every tame invariant 𝔵, if 𝔵 < 𝔶 holds in some forcing extension, then it holds in the P_{𝔶} extension.

Tame invariants are defined in B.14. Thus the forcing P_{𝔶} can be understood as increasing the invariant 𝔶 in the gentlest way, leaving all tame invariants smaller than 𝔶 if at all possible. Of course, a definition of this kind calls for a theorem to show that the defined concept is not empty.

Theorem 7.2. [39] (ZFC+LC) The following invariants can be isolated.

• 𝔟; P_{𝔟} is the iterated Laver forcing

• 𝔠; P_{𝔠} is the iterated Sacks forcing

• 𝔡; P_{𝔡} is the iterated Miller forcing

• cov(meager); P_{cov(meager)} is the iterated Cohen forcing

• cov(null); P_{cov(null)} is the iterated Solovay forcing.

All the items use the countable support iteration of length 𝔠^{+}.

This theorem shows that the correspondence between various invariants and forcing notions, for decades just an empirical fact of life, is no accident after all. It also has some immediate corollaries confirming some long-standing intuitions of the workers in this field:

Corollary 7.3. (Stated for 𝔠 only.)

• Whenever 𝔵 is a tame invariant, if ZFC+LC+𝔵 < 𝔠 is consistent then so is ZFC+LC+ℵ_{1} = 𝔵 < 𝔠 = ℵ_{2}.

• (ZFC+LC) Whenever 𝔵_{0}, … 𝔵_{n} are tame invariants and each of the inequalities 𝔵_{i} < 𝔠 holds in some forcing extension, then even the conjunction of these inequalities holds in some extension. Restated imprecisely, 𝔠 cannot be written as a nontrivial maximum of several tame invariants.

There are many other cardinal invariants that can be isolated. Let me just quote 𝔥, which is isolated by the iterated Mathias forcing. Also, the minimum of any combination of the invariants from Theorem 7.2 can be isolated by an iteration in which the relevant forcings alternate. There are invariants which cannot be isolated. A good example is cof(meager) since it can be written as max(𝔡, non(meager)). Both of the inequalities 𝔡 < cof(meager), non(meager) < cof(meager) are consistent [31], 2.2.11, 7.6.12, 7.5.8. Thus there is a fine line which divides the invariants into two areas: those which can be isolated and those which cannot. It seems to be of general interest to determine for every known invariant where it falls.

The reader should notice the at the moment somewhat mysterious duality between Theorem 7.2 and the P_{max} results of [36], [22]. Where the theorem proves the mutual consistency of inequalities 𝔵 < 𝔟, the P_{max} variation proves the mutual consistency of inequalities ℵ_{1} = 𝔡 < 𝔵. Comparing the two, we see that the P_{max} variation can treat a syntactically larger class of sentences, however the inequalities are always tied to ℵ_{1}. At the same time the P_{max} technology uses forcing extensions of models of determinacy while the theorems in this paper use extensions of an arbitrary universe of ZFC+LC.

All the results stated in Theorem 7.2 follow from the following general fact and the results of Subsection 3. Note that 𝔠 is the covering number of the ideal of the countable sets, 𝔟 is the covering number of the ideal associated with the Laver forcing and 𝔡 is the covering number of the ideal of σ-bounded sets, associated with the Miller forcing.

Theorem 7.4. (ZFC+LC) Suppose that I is a projectively generated projectively defined ideal such that

1. ZFC+LC proves that P_{I} is proper

2. ZF+AD proves that every set of reals either belongs to I or it has a Borel I-positive subset

3. I is homogeneous.

For every tame invariant 𝔵, if 𝔵 < cov(I) holds in some forcing extension, then it holds in the iterated P_{I}-extension.

If the assumption on the homogeneity of the ideal I is dropped, the invariant cov(I) must be replaced by the less appetizing min{cov(I ↾ B) : B ⊂ ℝ is a Borel I-positive set}.

Proof. I will prove the theorem under the additional assumption of CH. If CH fails, just observe that after the first ω_{1} steps of the iteration CH is restored and repeat the argument in that model.

Fix the ideal I with the properties enunciated in the theorem. Fix an arbitrary tame invariant 𝔵, the minimum size of a set A of reals with properties ϕ(A) and ∀x ∈ ℝ ∃y ∈ A θ(x, y) where ϕ quantifies over natural numbers and elements of A only and θ is a projective formula making no mention of the set A. I have to show that if 𝔵 < cov(I) holds in some forcing extension then it holds in the iterated P_{I} extension. The key ingredient of the argument is the isolation of a projective formula ψ(A) such that

1. (ZFC+LC) ψ(A) implies that ϕ(A) and ∀x ∈ ℝ ∃y ∈ A θ(x, y) holds after the countable support iteration of the poset P_{I} of arbitrary length 2. (ZFC+LC) If A is a set of reals witnessing 𝔵 < cov(I) then ψ(A) holds.

The proof of the theorem will then be concluded by a simple Σ_{1}^{2} absoluteness argument. Namely, note that ∃A ⊂ ℝ ψ(A) is a Σ_{1}^{2} statement. If 𝔵 < cov(I) holds in some forcing extension, then by (2) above ∃A ⊂ ℝ ψ(A) holds in that extension. By Fact C.6, ∃A ⊂ ℝ ψ(A) holds in the ground model, which is assumed to satisfy CH. Fix such a set A. By (1) above, in the iterated P_{I} model the set A will witness that ℵ_{1} = 𝔵 < cov(I) = ℵ_{2} as desired.

Now the formula ψ(A) is defined to say the following: ϕ(A) holds and for every countable ordinal α, every Borel I-perfect set B ⊂ ℝ^{α} and every Borel function f : B → ℝ there is a real y ∈ A such that the set {r⃗ ∈ B : θ(f(r⃗), y)} contains a Borel I-perfect subset. Clearly, ψ(A) is a projective property of the set A ⊂ ℝ. It is not hard to see that ψ(A) implies ϕ(A) ∧ ∀x ∈ ℝ ∃y ∈ A θ(x, y), in other words that A is a possible witness for the invariant 𝔵. In fact ψ(A) implies that A is a very stable such witness.

To verify the first item above, let A ⊂ ℝ be a set with the property ψ. It must be proved that the countable support iteration P_{I}^{β} of the poset P_{I} of an arbitrary ordinal length β preserves the property ϕ(A) ∧ ∀x ∈ ℝ ∃y ∈ A θ(x, y). The first conjunct, ϕ(A), is clearly preserved due to its simple syntactical form. For the second conjunct, suppose that ẋ is a P_{I}^{β}-name for a real. By Corollary 4.8, there is a condition B ∈ P_{I}^{β} with domain X ∈ [β]^{ℵ_{0}}–so B ⊂ ℝ^{X} can be assumed an I-perfect set–and a Borel function f : B → ℝ such that B ⊩ ẋ = ḟ(r⃗_{gen} ↾ X̌) where r⃗_{gen} is the name for the generic sequence of reals. Use ψ(A) with α = ot(X) to find a real y ∈ A such that the set {r⃗ ∈ B : θ(f(r⃗), y)} contains a Borel I-perfect subset C. By the projective absoluteness C.4 the condition C forces θ(ẋ, y̌) as desired. In fact, it is possible to prove that ψ(A) itself is preserved under the iteration.

To verify the second item work in the theory ZFC+LC+𝔵 < cov(I) and fix a witness A ⊂ ℝ for 𝔵 such that |A| < cov(I). To prove that the set A has the property ψ, fix a countable ordinal α, a Borel I-perfect set B ⊂ ℝ^{α} and a Borel function f : B → ℝ. By Corollary 4.16 one of the < cov(I)-many projective sets B_{y} = {r⃗ ∈ B : θ(f(r⃗), y)}, y ∈ A covering the set B, must have an I-perfect Borel subset!

Ciesielski and Pawlikowski in [13] formulated the axiom CPA which holds in the Sacks model and is intended to capture the combinatorial features of the model. A natural question can be asked: is CPA strong enough to imply all the possible sentences 𝔵 < 𝔠 that hold in the Sacks model by the previous theorem? It turns out that the answer is positive:

Theorem 7.5. (ZFC+LC) Suppose that 𝔵 is a tame invariant and 𝔵 < 𝔠 holds in some forcing extension. Then 𝔵 < 𝔠 holds in every forcing extension satisfying CPA.

Roughly restated, if a sentence of the form 𝔵 < 𝔠 is consistent, then it should follow from CPA. In particular, if ZFC+LC proves that 𝔵 < 𝔠 can be forced, then ZFC+LC+CPA proves that 𝔵 < 𝔠 holds. This should be compared to the Σ_{1}^{2} absoluteness theorem C.6 of Woodin, which roughly says that every consistent Σ_{1}^{2} sentence is implied by the Continuum Hypothesis. There is a notable difference between these two: the Continuum Hypothesis is itself a Σ_{1}^{2} statement, while CPA is substantially more complex than the statements of the form 𝔵 < 𝔠. There are parallel axiomatizations of the Miller model or the Laver model which satisfy the corresponding version of 7.5.

Proof. The axiom CPA in its full form is quite complicated. I will use just its weaker and more natural form, called CPA_{prism}^{game} in [13]. Consider the following two person game G of ω_{1} many rounds: in round β ∈ ω_{1} Adam chooses a countable ordinal α_{β}, a Borel ctble-perfect set B_{β} ⊂ ℝ^{α_{β}} and a Borel function f_{β} : B_{β} → ℝ. Eve then chooses a Borel I-perfect set C_{β} ⊂ B_{β}. In the end, Adam wins if ℝ = ⋃_{β∈ω_{1}} f′′_{β}C_{β}. CPA_{prism}^{game} then is the statement “𝔠 > ℵ_{1} and Eve has no winning strategy in the game G”.

Now let 𝔵 be a tame invariant, defined as the minimum size of a set A of reals with properties ϕ(A) and ∀x ∈ ℝ ∃y ∈ A θ(x, y) where ϕ quantifies over natural numbers and elements of A only and θ is a projective formula making no mention of the set A. Suppose that 𝔵 < 𝔠 holds in some generic extension; I will show that in an arbitrary forcing extension V [G] satisfying CPA 𝔵 = ℵ_{1} holds.

Move into the model V [G] and consider the formula ψ(A) from the previous proof. The proof shows that ∃A ⊂ ℝ ψ(A) holds in an arbitrary forcing extension satisfying CH. In particular, writing P for the canonical σ-closed forcing collapsing 𝔠 to ℵ_{1}, we have P ⊩ ∃A ψ(A). Fix P -names Ȧ and {ẏ_{γ} : γ ∈ ω̌_{1}} such that P ⊩ ψ(Ȧ) and Ȧ = {ẏ_{γ} : γ ∈ ω̌_{1}}. Since the formula ϕ(A) from the definition of the invariant quantifies over natural numbers and elements of A only, it is not difficult to produce open dense subsets {O_{β} : β ∈ ω_{1}} such that for every filter H ⊂ P meeting all of them the set A = {r ∈ ℝ : for some p ∈ H and γ ∈ ω_{1}, p ⊩ y̌ = ẏ_{γ}} has the property ϕ(A).

Consider the following strategy σ for Eve in the game G. She produces Borel ctble-perfect sets C_{β} ⊂ B_{β} and on the side, a descending chain ⟨p_{β} : β ∈ ω_{1}⟩ of conditions in the poset P such that for every β there is a real y and a countable ordinal γ such that for every sequence r⃗ ∈ C_{β}, θ(f_{β}(r⃗), y) holds, p_{β} ∈ O_{β} and p_{β} ⊩ y̌ = ẏ_{γ}. This is possible by the choice of the formula ψ. By CPA_{prism}^{game}, the strategy σ is not winning for Eve, and there must be a counterplay against it such that in the end Adam wins, meaning ℝ = ⋃_{β} f′′_{β}C_{β}. Let ⟨p_{β} : β ∈ ω_{1}⟩ be the conditions that Eve recorded in this run, and set A = {y ∈ ℝ :for some countable ordinals β and γ, p_{β} ⊩ y̌ = ẏ_{γ}}. By the choice of the open dense sets {O_{β} : β ∈ ω_{1}}, ϕ(A) holds. Since Adam won the run of the game, for every x ∈ ℝ there is y ∈ A such that θ(x, y) holds. Clearly, |A| = ℵ_{1}, therefore the set A is a witness to 𝔵 = ℵ_{1} as desired.

In the special case of the Sacks model (and CPA as its axiomatization) it is possible to prove a somewhat stronger theorem. Say that 𝔵(F) is a tame ultrafilter invariant if:

1. the syntax of its definition is the same as with tame invariants, except that an extra predicate F for a subset of P(ω) can be used in both formulas ϕ and θ

2. the predicate F stands for an ultrafilter on ω

3. if F is an ultrafilter, the value of the cardinal 𝔵(F) is invariant under permutations of F. That is, ZFC proves that whenever π : ω → ω is a permutation and F is an ultrafilter then 𝔵(F) = 𝔵(π(F)).

Theorem 7.6. (ZFC+LC) Suppose that 𝔵(F) is a tame ultrafilter invariant. If there is a forcing extension in which there is a Ramsey ultrafilter F such that 𝔵(F) < 𝔠, then in every forcing extension satisfying CPA, for all Ramsey ultrafilters F we have 𝔵(F) = ℵ_{1} < 𝔠.

Thus CPA implies that all Ramsey ultrafilters have the same cardinal characteristics, and these are equal to ℵ_{1} if at all possible. A necessary remark at this point is that CPA implies the existence of Ramsey ultrafilters by the work of [13] or by the previous theorem.

Proof. This is very similar to the proof of the previous theorem, replacing the references to 4.14 by 4.18 and references to the Σ_{1}^{2} absoluteness C.6 by its variation parametrized by Ramsey ultrafilters in [40]. To be specific, suppose that 𝔵(F) is a tame ultrafilter invariant, defined as the minimum size of a set A of reals with properties ϕ(A, F) and ∀x ∈ ℝ ∃y ∈ A θ(x, y, F) where ϕ quantifies over natural numbers and elements of A only and θ is a projective formula making no mention of the set A. Suppose that in some generic extension V [H] there is a Ramsey ultrafilter F such that 𝔵(F) < 𝔠 holds. The argument in the proof of Theorem 7.4 together with 4.18 and the invariance of 𝔵(F) under permutations of the ultrafilter F shows that in V [H] the following statement holds about F: ∀π : ω → ω, a permutation, ∃A ⊂ ℝ such that ψ(A, πF), where ψ(A, πF) = ϕ(A, πF)∧∀α ∈ ω_{1}∀B ⊂ ℝ^{α} Borel ctble-perfect ∀f : B → ℝ Borel ∃y ∈ A such that the set {r⃗ ∈ B : θ(f(r⃗), y, F)} contains a Borel ctble-perfect subset. This statement about the ultrafilter F is invariant under permutations of F, and it is easy to restate as a Σ_{1}^{2}(F) statement. Thus by the work of [40] it holds in all generic extensions satisfying CH about all of their Ramsey ultrafilters. The rest of the argument is identical to the previous proof, and it is left to the reader.

7.2 Optimality of the countable support iteration

Theorem 7.4 is very useful in dealing with an ideal which is fairly familiar and has quite special properties. However the situation is frequently not that simple. In such a case it is necessary to settle for somewhat weaker conclusions, which can still carry the desired methodological impact.

Theorem 7.7. (ZFC+LC) Assume that the Continuum Hypothesis holds. Suppose that I is a projectively generated projectively defined ideal such that 1. ZFC+LC proves that P_{I} is proper

2. I is homogeneous.

Whenever 𝔵 is a tame cardinal invariant such that 𝔵 < cov(I) holds in some proper forcing extension then it holds in the iterated P_{I}-extension.

Thus compared to Theorem 7.4 the AD dichotomy assumption on the ideal is dropped, and the price paid for this is that instead of all generic extensions we can look only at proper forcing extensions of models of the Continuum Hypothesis. The theorem says that the countable support iteration of the poset P_{I} has all the preservation properties a proper forcing increasing cov(I) can possibly have. It is applicable for ideals such as the ideal of E_{0}-small sets of Subsection 3.6.

The proof of Theorem 7.7 closely follows the argument for 7.4, with Corollary 4.24 replacing the reference to Corollary 4.16. Use Facts C.5 and C.8 to observe that in proper forcing extensions of a ZFC+LC+CH universe the cardinal 𝔡𝔢𝔱 alluded to in 4.24 is equal to ℵ_{1}.

Theorem 7.8. (ZFC+LC) Suppose that P is a provably proper definable real forcing and 𝔵 is a tame invariant. If 𝔵 = ℵ_{1} holds together with the forcing axiom for P in some generic extension, then it holds in the countable support iterated P -extension.

The forcing axiom for P is the statement that for every condition p ∈ P and every collection of ℵ_{1} open dense subsets of the poset P, there is a filter containing the condition p and meeting every one of the open dense sets. The forcing axiom for P holds after iterating P with any reasonable notion of support (and it is also possible to force without iterating P –see [33]). Theorem 7.8 then says that the countable support iteration of the poset P has all preservation properties a decent iteration of P can possibly have.

Clearly, Theorem 7.8 is useful in situations where the emphasis is more on the forcing and less on the ideal, such as the case may be with say Silver forcing. The price paid for neglecting the relevant ideal is the necessity to consider the forcing axiom instead of the covering number, a strictly stronger and a strictly more complex statement.

The argument for Theorem 7.8 is essentially identical to that of Theorem 7.4 using the analysis of the countable support iteration of the poset P obtained in Section 2, replacing Lemma 4.10 with 4.23. One difference is that the forcing P_{I(P)} is not necessarily a projective collection of Borel sets in this general case, but still it is universally Baire, being obtained from P using the game quantifier. As a result, the formula ψ from the proof of Theorem 7.4 is no longer projective– it will use the universally Baire predicate for the Borel part of I(P). However, Theorem C.6 applies to sentences with such parameters, and the argument will go through.

7.3 Duality

The following theorem is a partial justification of the so-called duality heuristic that is frequently used to prove inequalities between cardinal invariants. The heuristic is explained in [31]; very roughly, it says “if the covering number is provably large then the uniformity number must be provably small”.

Theorem 7.9. [37] Let I be an arbitrary projectively generated projectively defined σ-ideal. If ZFC+LC proves cov(I) = 𝔠 then ZFC+LC proves non(I) < ℵ_{4}.

The large cardinal hypothesis in the conclusion of this theorem must be stronger than both the large cardinal hypothesis in the assumption and “ω_{1} many Woodin cardinals”. However, I do not know one single interesting case of a σ-ideal I for which the large cardinal hypothesis would make any difference. The theorem stays true if the pair cov, non is replaced by any other dual pair of invariants.

Proof. The key ingredient in the argument is the computation of the cardinal coefficients of the Fubini powers of the countable ideal in Lemma 4.11. Suppose that I is a projectively generated projectively defined ideal such that ZFC+LC proves cov(I) = 𝔠. Then ZFC+LC proves also the sentence ψ saying that there is a countable ordinal α, a Borel ctble-perfect set B ⊂ ℝ^{α} and a Borel function f : B → ℝ such that the f-preimages of I-small sets do not have Borel ctbleperfect subsets (which is to say they are ctble^{α}-small). For if this were not the case, in the consistent theory ZFC+LC+¬ψ Corollary 4.8 could be used to conclude that in the iterated Sacks extension cov(I) = ℵ_{1} < 𝔠 contradicting the assumptions on the ideal I. Now within the theory ZFC+LC use ψ and Lemmas 4.11 and 4.17 to conclude that non(I) < ℵ_{4}: given a countable ordinal α, a Borel ctble-perfect set B ⊂ ℝ^{α}, a Borel function f : B → ℝ such that the f-preimages of I-small sets are ctble^{α}-small (from ψ), a ctble^{α}-positive set A ⊂ ℝ^{α} of size < ℵ_{4} (from Lemma 4.11), and a function g : ℝ^{α} → B such that the g-preimages of ctble^{α}-small sets are ctble^{α}-small (Corollary 4.17), the set (f ∘ g)′′A ⊂ ℝ must be I-positive of size < ℵ_{4} as desired.

Very recently Shelah (personal communication) proved a form of club guessing on ℵ_{2} which makes it possible to lower the cardinal ℵ_{4} in the above theorem to ℵ_{3}. However, it cannot be lowered to ℵ_{2}, as the following example shows.

Example 7.10. Let J be the σ-ideal on ℝ^{ω} σ-generated by the sets A_{g} = {r⃗ ∈ ℝ^{ω} : ∃i ∈ j ∈ ω r⃗(j) = g(r⃗(i))} as g varies over all Borel functions from ℝ to ℝ. Then ZFC proves cov(J) = 𝔠 and ZFC+PFA proves non(J) = ℵ_{2}.

Proof. If {g_{γ} : γ ∈ κ} are < 𝔠 many Borel functions from ℝ to ℝ, by induction on j ∈ ω find reals r⃗(i) so that r⃗(i) is not in the set {g_{γ}(r⃗(j)) : j ∈ i, γ ∈ κ} of size < 𝔠. Clearly the sequence r⃗ ∈ ℝ^{ω} is not in any of the sets A_{g_{γ}} showing that cov(J) = 𝔠.

Now work in the context of PFA and fix a set A ⊂ ℝ^{ω} of size ℵ_{1}. I will show that A ∈ J, and therefore non(J) = ℵ_{2}. Let r_{α} : α ∈ ω_{1} be an enumeration of the set X of all the reals occuring on the sequences in the set A. By an application of PFA to the Baumgartner’s poset adding a club with finite conditions, there is a club C ⊂ ω_{1} such that for every sequence r⃗ ∈ A there are integers i ∈ j and two successive elements of the club C such that the reals r⃗(i), r⃗(j) were enumerated between these. Let f_{n} : X → X, for n ∈ ω be functions such that for every ordinal α ∈ ω_{1} the functional values f_{n}(r_{α}) : n ∈ ω include every real enumerated before the least element of the club C larger than α. By an application of PFA to the c.c.c. coding poset of Claim B.13, there are Borel functions g_{n} : n ∈ ω such that for every integer n f_{n} ⊂ g_{n}. It is immediately clear that A ⊂ ⋃_{n} A_{g_{n}}.

Other duality theorems are possible:

Theorem 7.11. Suppose that I is a projectively generated projectively defined ideal such that ZFC+LC proves that 𝔡 ≤ cov(I). Then ZFC+LC proves that 𝔟 ≥ ℵ_{2} implies non(I) ≤ 𝔟.

7.4 Interpolation theorems

Interpolation theorems in logic assert that if an implication ϕ → ψ is provable, then there must be an interpolant θ such that both implications ϕ → θ and θ → ψ are provable. Reviewing the methodology of various fields of mathematics, it is possible to point out many interpolation heuristics with very natural interpolants. The challenge is to support these heuristics with theorems. In this subsection I will present several such theorems. To my knowledge these are the first interpolation theorems for strong theories. As a matter of notation, let the diagram

[figure: ϕ; ψ; θ]

denote the following statement: ZFC+LC proves θ → ψ, and if ZFC+LC proves ϕ → ψ then ZFC+LC proves ϕ → θ. This will be invariably used in a situation where the sentences θ and ψ are fixed and the proof of θ → ψ is simple and long known. On the other hand, the sentence ϕ will vary within a wide syntactically identified class of statements, and θ will belong to that class. The moral captured in the diagram is then this: every argument similar to θ → ψ must go through θ → ψ, in other words θ → ψ is the critical argument of its kind. This must be illustrated on examples.

A Lusin set is an uncountable set of reals which has a countable intersection with every meager set. Lusin sets provide topologists with a rich source of examples. Let LS denote the sentence “there is a Lusin set”. LS is a Σ_{1}^{2} sentence independent of ZFC. Not surprisingly, the Continuum Hypothesis implies LS, and that argument can be easily refined to give cof(meager) = ℵ_{1} →LS.

Theorem 7.12. Let 𝔵 be an arbitrary tame cardinal invariant. Then

[figure: 𝔵 = ℵ_{1}; LS; cof(meager) = ℵ_{1}]

Proof. Suppose that ZFC+LC does not prove 𝔵 = ℵ_{1} → cof(meager) = ℵ_{1}. So there must be a countable model M of the consistent theory ZFC+LC+𝔵 = ℵ_{1}+cof(meager) > ℵ_{1}. Since ZFC proves that cof(meager) = max{𝔡, non(meager)}, there are two cases. Either M ⊧ non(meager) > ℵ_{1}. But in such a case the model M trivially does not have any Lusin sets, and therefore shows that ZFC+LC does not prove 𝔵 = ℵ_{1} →LS. Else, it must be the case that M ⊧ 𝔡 > ℵ_{1} = 𝔵. Then look into the iterated Miller extension M[G] of M. By Theorem 7.2 applied in the model M, M[G] ⊧ 𝔵 = ℵ_{1} and by an old result about the Miller forcing [22], 6.7, M[G] has no Lusin sets. Therefore the model M[G] stands witness to the ZFC+LC unprovability of the implication 𝔵 = ℵ_{1} →LS. The theorem follows.

The Borel Conjecture (BC) is the statement that every strong measure zero set of reals is countable, where a set A of reals is strong measure zero if for every function f ∈ ω^{ω} there exist intervals I_{n} : n ∈ ω such that the length of I_{n} is 2^{−f(n)} and the set A is covered by their union–[31] Chapter 8.

Theorem 7.13. Let 𝔵 be an arbitrary tame cardinal invariant. Then

[figure: 𝔵 = ℵ_{1}; ¬BC; 𝔟 = ℵ_{1}]

Proof. 𝔟 = ℵ_{1} → ¬BC is proved in [31], Theorem 8.2.8. Suppose now that 𝔵 = ℵ_{1} → 𝔟 = ℵ_{1} is not provable. Thus the theory ZFC+LC+𝔵 < 𝔟 is consistent and realized in some countable model M. By Theorem 7.2, in the iterated Laver extension M[G] of that model, 𝔵 = ℵ_{1} < ℵ_{2} = 𝔟 holds. By a celebrated theorem of Laver [14], Borel conjecture holds in M[G]. Thus the model M[G] witnesses that the implication 𝔵 = ℵ_{1} → ¬BC can consistently fail.

Question 7.14. Let DBC denote the Dual Borel Conjecture–[31] Section 8.5. Bartoszynski recently proved that 𝔡 = ℵ_{1} implies the failure of DBC. Does the following hold for every tame invariant 𝔵?

[figure: 𝔵 = ℵ_{1}; ¬DBC; 𝔡 = ℵ_{1}]

This is really a question on whether DBC holds in the Miller model.

A good subject for interpolation theorems is the relationship between the tame invariants and prediction principles at ω_{1}. There the interpolants are usually clearly visible to the naked eye. However, the models used in the proofs must be different than those in Theorem 7.2 because that theorem uses models which satisfy very strong prediction principles. Let me mention two examples.

The Weak Club Guessing Principle WCG is the statement that there is a collection of size ℵ_{1} consisting of subsets of ω_{1} of ordertype ω such that every closed unbounded subset of ω_{1} has infinite intersection with one of them.

Theorem 7.15. For every tame invariant 𝔵,

[figure: 𝔵 = ℵ_{1}; WCG; 𝔟 = ℵ_{1}]

Proof. The argument uses the ideals J_{α} and their tower P_{S} introduced in Subsection 6.2. The implication 𝔟 = ℵ_{1} →WCG follows immediately from Lemma 6.18.

Now choose a tame invariant 𝔵 = min{|A| : A ⊂ ℝ, ϕ(A) ∧ ∀x ∈ ℝ∃y ∈ A θ(x, y)} and suppose that ZFC+LC does not prove 𝔵 = ℵ_{1} → 𝔠 = ℵ_{1}; it must be shown that then it does not prove 𝔵 = ℵ_{1} →CG either. There is a countable model M of the consistent theory ZFC+LC+𝔵 = ℵ_{1} < 𝔟 and a witness A ⊂ ℝ in it for 𝔵 = ℵ_{1}. Look at the iterated P_{S}-extension M[G] of M. By a chain condition argument and Lemma 6.17 M[G] ⊧ ¬WCG. Thus the proof will be complete if I show that the set A remains a witness for 𝔵 = ℵ_{1} in the model M[G] because then M[G] stands witness to the consistency of ZFC+LC+𝔵 = ℵ_{1}+¬WCG.

Move to the model M. To see that A will be a witness in the extension, let B be a condition in the iteration and ẋ a name for a real. A stronger condition C ≤ B and a real y ∈ A must be found such that C ⊩ θ(ẋ, y̌). Using Lemma 6.7, strengthening the condition B if necessary we may assume that there is an indecomposable countable ordinal α, a countable set X of ordinals and a function f such that B ⊂ (𝔽_{α})^{X} is a Borel J_{α}-perfect set and f : B → ℝ is a Borel function such that B ⊩ ẋ = ḟ(F⃗_{gen} ↾ (dom(B) × α)). Look at the sets C_{y} = {g⃗ ∈ B : θ(f(g⃗), y)} for y ∈ A. These are < 𝔟 many projective sets covering the set B, so by Corollary 6.19 one of them, say C_{y}, must have a Borel J_{α}-perfect subset C ≤ B which will then force θ(ẋ, y̌) by Lemma 6.7 and the projective absoluteness C.4. The theorem follows.

It is typical that the proof of 𝔟 = ℵ_{1} →WCG is quite easy and one gets a much stronger conclusion: actually every infinite subset of ω_{1} can be predicted.

The Club Guessing Principle CG is the statement that there is a collection of size ℵ_{1} consisting of infinite subsets of ω_{1} such that every closed unbounded subset of ω_{1} contains one of them as a subset.

Theorem 7.16. For every tame invariant 𝔵,

[figure: 𝔵 = ℵ_{1}; CG; 𝔠 = ℵ_{1}]

Proof. The proof of 𝔠 = ℵ_{1} →CG is trivial: the collection of all bounded subsets of ω_{1} is of size 𝔠 and it predicts even every infinite subset of ω_{1}. The rest of the argument follows the familiar pattern using the ideals I_{α} and their tower P_{T} introduced in Subsection 6.1. Choose a tame invariant 𝔵 = min{|A| : A ⊂ ℝ, ϕ(A) ∧ ∀x ∈ ℝ∃y ∈ A θ(x, y)} and suppose that ZFC+LC does not prove 𝔵 = ℵ_{1} → 𝔠 = ℵ_{1}; it must be shown that then it does not prove 𝔵 = ℵ_{1} →CG either. Let M be a model of the consistent theory ZFC+LC+𝔵 < 𝔠 and let M[G] be its iterated P_{T} extension. Lemma 6.11 plus a chain condition argument show that in the model M[G] the Club Guessing fails. The proof will be complete if I show that M[G] ⊧ 𝔵 = ℵ_{1}.

Move into the model M and pick a witness A ⊂ ℝ for the invariant 𝔵 such that |A| < 𝔠. I will argue that the countable support iteration of P_{T} forces that A is still a witness for 𝔵. This will be enough since in the extension |A| ≤ |𝔠|^{M} = ℵ_{1}. To see that A will be a witness in the extension, let B be a condition in the iteration and ẋ a name for a real. A stronger condition C ≤ B and a real y ∈ A must be found such that C ⊩ θ(ẋ, y̌). Using Lemma 6.7, strengthening the condition B if necessary we may assume that there is an indecomposable countable ordinal α, a countable set X of ordinals and a function f such that B ⊂ (𝔽_{α})^{X} is a Borel I_{α}-perfect set and f : B → ℝ is a Borel function such that B ⊩ ẋ = ḟ(F⃗_{gen} ↾ dom(B) × α)). Look at the sets C_{y} = {g⃗ ∈ B : θ(f(g⃗), y)} for y ∈ A. These are < 𝔠 many projective sets covering the set B, so by Lemma 6.12 one of them, say C_{y}, must have a Borel I_{α}-perfect subset C ≤ B which will then force θ(ẋ, y̌) by Lemma 6.7 and the projective absoluteness C.4. The theorem follows.

There are many similar theorems that can be produced using the tower technology. Let me state two more results without proof. Let Q denote the statement that there is a collection of ℵ_{1} many subsets of ω_{1} of ordertype ω such that for every partition of ω_{1} into countably many pieces one of the pieces has infinite intersection with one set in the collection.

Theorem 7.17. For every tame invariant 𝔵,

[figure: 𝔵 = ℵ_{1}; Q; 𝔟 = ℵ_{1}]

Let P denote the statement that there is a collection of ℵ_{1} many infinite subsets of ω_{1} such that for every partition of ω_{1} into countably many pieces one of the pieces contains one of the sets in the collection as a subset.

Theorem 7.18. For every tame invariant 𝔵,

[figure: 𝔵 = ℵ_{1}; P; 𝔠 = ℵ_{1}]

It seems to be of a general interest to determine which prediction principles on ω_{1} are such that their impact on tame invariants can be isolated in one invariant. Let me finish with an apparently hard open question.

Question 7.19. By an old result of Ketonen [31] Theorem 4.4.5, 𝔡 = 𝔠 implies the existence of a P -point. Is the following true for every covering number 𝔵:

[figure: 𝔵 = 𝔠; P -point; 𝔡 = 𝔠]

The solution to the question probably has to address the problem whether every two consistently unequal covering numbers can be separated so that one of them is ℵ_{1} and the other ℵ_{2}.

A Some cardinal characteristics of the continuum

𝔞 The smallest size of a maximal almost disjoint family. A family A of infinite subsets of ω is almost disjoint if its elements have finite intersection. 𝔞 is tame, but it is not a covering number. 𝔟 ≤ 𝔞 holds in ZFC and 𝔟 < 𝔞 is consistent by [26], Section VI.6. The consistency of 𝔡 < 𝔞 was an open problem for a long time, solved by [23]. It is still unknown whether ℵ_{1} = 𝔡 < 𝔞 is consistent.

𝔟 The smallest size of an unbounded family. A family A of functions from ω to ω is unbounded if there is no single function f such that every function g ∈ A is smaller than f except for a finite number of values. 𝔟 is a covering number. 𝔟 can be isolated by the iterated Laver forcing and there is an optimal P_{max} style model for 𝔟 = ℵ_{1}.

𝔠 The size of the continuum. It is a covering number. It can be isolated by the iterated Sacks forcing.

𝔡 The smallest size of a dominating family. A family A of functions from ω to ω is dominating if for every function f there is g ∈ A which is larger than f except for a finite number of values. 𝔡 is a covering number. 𝔡 can be isolated by the iterated Miller forcing and there is an optimal P_{max} style model for 𝔡 = ℵ_{1}.

𝔤 The smallest size of a collection of groupwise dense families with empty intersection. A family H ⊂ [ω]^{ℵ_{0}} is groupwise dense if it is open dense in the algebra P(ω) \ I where I is the ideal of finite sets, and moreover for every partition {I_{n} : n ∈ ω} of ω into finite sets there is an infinite set a ⊂ ω such that ⋃_{n∈a} I_{n} ∈ H. In ZFC, 𝔱 ≤ 𝔥 ≤ 𝔤 ≤ 𝔡 and none of these inequalities can be reversed.

𝔥 The distributivity of the algebra P(ω) \ I where I is the ideal of finite sets. Restated, it is the smallest size of a collection of open dense subsets of that algebra with an empty intersection. 𝔥 is not tame. It can be isolated at ℵ_{2} by the iterated Mathias forcing. 𝔥 ≤ 𝔟 and 𝔥 ≤ 𝔰 and these inequalities cannot be reversed.

𝔱 The tower number, the smallest size of a tower without a bound. A tower is a function f : α → [ω]^{ℵ_{0}} for some ordinal α such that γ ∈ β ∈ α implies f(β) ⊂ f(γ) except for a finite number of elements. A bound for such a tower f is an infinite set a ⊂ ω which is up to a finite number of elements contained in every set of the range of the tower f.

𝔷 The coding number, the smallest size of a partial function from ℝ to ℝ which is not a subset of a Borel function. This is not one of the standard invariants, but it does occur naturally in several places in the text. It is not a covering number.

B The syntax of cardinal characteristics of the continuum

In this section I analyze in some detail the syntax of the definition of various cardinal invariants. There seem to be at least three syntactically distinguishable categories of invariants: the covering numbers, the tame invariants and general invariants. I will define the first two and give several examples. While the statements of Examples B.8, B.11, B.16 and B.17 are interesting from this standpoint, their proofs are completely irrelevant to the rest of the paper and can be skipped.

B.1 The covering numbers

Definition B.1. A σ-ideal I on the reals is called projectively generated projectively defined if there are integers n, m such that every set in I is a subset of a Σ_{n}^{1} set in I and the set of all codes for Σ_{n}^{1} sets in I is Σ_{m}^{1}.

Several theorems in this paper requires the latter set to be actually ligthface Σ_{m}^{1} and its definition will be the definition of the ideal in question. The boldface versions of the theorems can be somewhat more complicated to state.

Example B.2. The ideal of all σ-bounded subsets of ω^{ω} is projectively generated and defined: it is σ-generated by the sets A_{g} = {f ∈ ω^{ω} : f < g everywhere} as g ranges over all functions in ω^{ω}.

Example B.3. The ideal I of all sets nowhere dense in the algebra P(ω) \ J, where J is the ideal of finite sets, is projectively defined, but not projectively generated. By this I mean that for a set A ⊂ P(ω) the statement A ∈ I is projective, however the ideal I is not generated by its projective elements. I do not see an elementary proof of the latter statement–see [41].

Definition B.4. A cardinal invariant 𝔵 is a covering number if there is a projectively generated projectively defined σ-ideal I such that 𝔵 is defined as the smallest size of a collection of sets in the ideal I which covers the whole real line.

Example B.5. cov(meager) is a covering number by definition.

Example B.6. 𝔡 is a covering number: it is the covering number of the ideal of σ-bounded sets.

Given a projectively defined projectively generated ideal J, the set theorists will study its cardinal invariants: its covering number cov(J), the additivity number add(J), which is the smallest size of a family of sets in the ideal whose union is not in the ideal; the uniformity number non(J), which is the smallest size of a J-positive set, and the cofinality number cof(J), which is the smallest size of a basis for the ideal J. All of these invariants can be restated as covering numbers of related ideals, as the following example shows. Generally, the intuition is that a covering number is the smallest possible size of an unorganized heap of real numbers which has a certain largeness property.

Example B.7. non(meager) is a covering number. To see this, consider the space X of countable sets of nowhere dense subtrees of 2^{<ω} and the ideal I on it σ-generated by the sets A_{r} = {p ∈ X : r is not a path through any of the trees in p} as r runs through all the elements of 2^{ω}. Clearly, cov(I) = non(meager).

Example B.8. 𝔱 cannot be written as a covering number.

Proof. To show this, I need a model M of ZFC+LC in which 𝔱 > ℵ_{1} and there is a Souslin tree. Such a model can be obtained from a model of diamond by iterating σ-centered forcings with finite support. In the model M I will use the following well-known facts:

Claim B.9. Every Souslin tree forces 𝔱 = ℵ_{1}.

Proof. Fix a Souslin tree T. A simple transfinite induction argument on the levels of the tree T produces a function F : T → [ω]^{ℵ_{0}} such that for arbitrary nodes s, t ∈ T,

• s < t if and only if F (s) ⊂ F (t) up to a finite number of elements

• s is incompatible with t if and only if F (s) ∩ F (t) is finite.

Let b be a generic branch through the tree T. The first item shows that F′′b is a modulo finite decreasing sequence of subsets of ω. Moreover, this sequence has no lower bound: such a lower bound a ⊂ ω would have to come from the ground model, but on the other hand, it would construct the generic branch b as b = {t ∈ T : a ∩ F (t) is infinite} by the second item above. Contradiction.

Claim B.10. Forcing with Souslin trees does not change the values of covering numbers.

Proof. Fix a Souslin tree T. Since the tree is ω-distributive it does not add any reals and no generating sets for any projectively generated projectively defined ideal. And since it has size ℵ_{1}, any uncountable set of reals in the generic extension (a candidate for a set of codes for a new covering collection) is covered by a set of the same size coming from the ground model.

So if 𝔱 could be written as a covering number, within the model M considering an arbitrary Souslin tree T one can see that forcing with T should both change the value of 𝔱 (Claim B.9) and leave the value of 𝔱 unchanged (Claim B.10). Contradiction.

Example B.11. 𝔞 cannot be written as a covering number.

Proof. This is substantially harder than the previous proof. The approach is the same: I will find a model of ZFC+LC+𝔞 > ℵ_{1} and a forcing in it which preserves ℵ_{1} as well as all cardinals and covering numbers, but collapses 𝔞 to ℵ_{1}. There are several models that can be used. I will use a result of Brendle [5]: every model N of CH has a c.c.c. extension M in which the N-reals are unbounded and 𝔞 = 𝔠 = ℵ_{2}.

Let P be the forcing in the model N defined so as to add a maximal almost disjoint family of size ℵ_{1} with countable approximations: the elements of P are functions whose domain is a countable ordinal and whose range consists of mutually almost disjoint infinite subsets of ω. The order is the reverse extension. Let me now consider the forcing P ∈ N as an element of the model M. Work in M. There are three items to be noted:

• The forcing P does not add any real numbers. It is a well-known fact that σ-closed forcings stay ω-distributive after c.c.c. forcing, and this fact applied in the model N to the poset P shows that P is still ω-distributive in the model M.

• The forcing P is of size ℵ_{1}. Putting this together with the first item, it is clear that the forcing P does not collapse cardinals or covering numbers.

• P forces that its generic almost disjoint family is maximal. Of course, this was clear in the model N, but the model M has some additional real numbers that could spoil the maximality. Well, suppose p ∈ P is a condition and a ∈ M is an infinite subset of ω. A strengthening q of p must be found whose range contains a set with infinite intersection with a. Rearranging the condition p to simplify the notation, I may assume that its domain is ω. Find a function f ∈ N ∩ ω^{ω} such that the sets p′(n) = p(n) \ f(n) : n ∈ ω are mutually disjoint. There are three cases:

1. The set a ∩ ⋃_{n} p′(n) is finite. Then the set b = ω \ ⋃_{n} p′(n) is in the model N, it has finite intersection with every set in the range of the condition p, and contains all elements of the set a up to possibly finitely many. The condition q = p ∪ {⟨ω, b⟩} is as required.

2. The set a ∩ ⋃_{n} p′(n) is infinite and for some number n, a ∩ p′(n) is infinite. Then a ∩ p(n) is infinite too and the condition q = p is as required.

3. The set a ∩ ⋃_{n} p′(n) is infinite, but for all numbers n ∈ ω, the set a ∩ p′(n) is finite. Since the reals of the model N are unbounded in M, there must be a function g ∈ N ∩ω^{ω} such that for infinitely many numbers n ∈ ω, a ∩ p′(n) is a nonempty set included in g(n) ∩ p′(n). Let b = ⋃_{n} g(n)∩p′(n): necessarily b ∈ N, b∩a is infinite while b∩p(n) is finite for every number n and the condition q = p ∪ {⟨ω, b⟩} is as required.

Thus within the model M the forcing P changes the size of 𝔞 but leaves all the covering numbers unchanged. Ergo, 𝔞 is not a covering number.

Example B.12. 𝔷 cannot be written as a covering number.

Proof. In fact, it should be the case that every covering number provably ≤ 𝔷 should be provably equal to ℵ_{1}. I do not know how to prove that. Instead I will show that if 𝔵 is a covering number and ZFC+LC proves 𝔵 ≤ 𝔷 then ZFC+LC+the nonstationary ideal is saturated proves 𝔵 < ℵ_{3}. This will show that 𝔷 is not a covering number, since it can be increased by a c.c.c. forcing which does not destroy the saturation:

Claim B.13. For every function f : ℝ → ℝ there is a c.c.c. partial ordering P introducing a Borel function ġ such that P ⊩ f̌ ⊂ ġ.

This is an elementary exercise in the almost disjoint coding technique [7] Section 26. View ℝ as the set of all branches through 2^{ω}, which then are almost disjoint subsets of 2^{<ω}. Then force sets a_{n} ⊂ 2^{<ω} such that for every n ∈ ω and every real r in the ground model, f(r)(n) = 0 if and only if |a_{n} ∩ r| = ℵ_{0}. In the extension, let g : ℝ → ℝ be defined by g(r)(n) = 0 if |a_{n} ∩ r| = ℵ_{0}.

Now suppose that ZFC+LC+saturation does not prove that 𝔵 < ℵ_{3} and move into a countable model M of the consistent theory ZFC+LC+saturation+𝔵 ≥ ℵ_{3}. Let G be an M-generic filter on the algebra of stationary subsets of ω_{1} and let j : M → N be the generic elementary embedding. The conclusion will follow from the comparison of the models N and M[G]:

• the models N and M[G] have the same reals.

• N covers M[G] in the sense that every set A ∈ M[G] of M[G]-size ℵ_{1} = ℵ_{2}^{M} of reals is a subset of a set B ∈ N of N-size ℵ_{2}. Since by the elementarity of the embedding j the model N satisfies 𝔵 ≥ ℵ_{3}, it follows that M[G] ⊧ 𝔵 ≥ ℵ_{2}. To see this, note that for the set A there is a ground model set F of size ℵ_{2}^{M} consisting of functions from ω_{1} to ℝ such that A = {jh(ω_{1}^{M}) : h ∈ F }. Then A is covered by the N-set B = {h(ω_{1}^{M}) : h ∈ j(F)} of N-size ℵ_{2}^{N}.

• The set j′′ω_{2}^{M} does not belong to the model N. Thus fixing a collection C = {r_{γ} : γ ∈ ω_{2}^{M}} of distinct reals in the model M, the function f : jC → 2 defined by f(r) = 1 if and only if r ∈ M cannot be a subset of any Borel function coded in N or M[G].

Thus in the model M[G] it is the case that 𝔷 = ℵ_{1} < 𝔵.

B.2 The tame invariants

Definition B.14. A cardinal invariant 𝔵 is tame if it is defined to be the minimum size of a set A of reals with properties ϕ(A) and ∀x ∈ ℝ ∃y ∈ A θ(x, y) where ϕ quantifies over natural numbers and elements of A only and θ is a projective formula making no mention of the set A. Real parameters are allowed. The set A is called a witness for 𝔵.

Example B.15. 𝔞 is tame. It is defined as the smallest size of a maximal almost disjoint family, so ϕ(A) will assert that the elements of A are mutually almost disjoint and θ(x, y) will assert that x ∩ y is infinite.

For a similar reason, invariants like 𝔰 and 𝔱 as well as all covering numbers are tame. The covering numbers can be written in such a way that the formula ϕ(A) is trivially true.

Example B.16. (independently proved by Mildenberger) 𝔥 is not tame.

Proof. Just like in the previous proofs, I will find a model M of ZFC+LC+𝔥 > ℵ_{1} and a forcing in it which preserves cardinals, preserves tame invariants and collapses 𝔥 to ℵ_{1}. The model M is the iterated Mathias model and the forcing P is simply the algebra P(ω) minus the finite sets. In the model M, 𝔠 = 𝔥 = ℵ_{2}, P is ℵ_{1}-distributive of size 𝔠 = ℵ_{2} and therefore does not collapse cardinals or tame invariants (why?) and by work of [27] it forces 𝔥 = ℵ_{1} as desired.

Example B.17. (Mildenberger) 𝔤 is not tame.

Proof. In fact I will prove that if 𝔵 is a tame invariant and ZFC+LC proves 𝔤 ≤ 𝔵 then ZFC+LC proves 𝔡 ≤ 𝔵, and 𝔤 ≤ 𝔡 holds by [31] Theorem 4.3.2. This will show that 𝔤 is not tame since 𝔤 < 𝔡 holds in the finite support iterated Cohen model by [31], 4.3.3.

So suppose that 𝔵 is a tame invariant such that ZFC+LC does not prove 𝔡 ≤ 𝔵. There must be a countable model M of the consistent theory ZFC+LC+𝔵 < 𝔡, and by Theorem 7.4, in its iterated Miller extension M[G], ℵ_{1} = 𝔵 < 𝔡 = ℵ_{2} holds. By [3] 𝔤 = 𝔡 = ℵ_{2} holds in the iterated Miller model as well. Thus the model M[G] stands witness to the consistency of the theory ZFC+LC+𝔵 < 𝔤.

The last note. If 𝔵 is a tame invariant then the statement 𝔵 = ℵ_{1} is Σ_{2}- expressible in the structure ⟨H_{ℵ_{2}}, ω_{1}, ∈⟩ and so the technology of P_{max} and its variations in [36], [22] applies to it. There appears to be a curious duality: a covering number cov(I) can be isolated (meaning that all consistent inequalities 𝔵 < cov(I) are mutually consistent for tame 𝔵) just in case there is an optimal P_{max} style model for non(I) = ℵ_{1} (meaning that all consistent inequalities ℵ_{1} = non(I) < 𝔵 are mutually consistent).

C Large cardinals

Definition C.1. [8], Section 26. A cardinal δ is Woodin if for every function f : δ → δ there is an ordinal α ∈ δ closed under the function and an elementary embedding j : V → M with a transitive target model, crit(j) = α and V_{j(α)} ⊂ M.

There is no need for the reader of this paper to understand the subtleties of the combinatorics of Woodin cardinals. All the necessary consequences are stated in this appendix section and can be understood and used as black boxes. In the hierarchy of large cardinals, Woodin cardinals are considerably stronger than measurable and strong cardinals and considerably weaker than supercompact or huge cardinals.

Definition C.2. [19] A class tree T ⊂ (Ord ×ω)^{<ω} is universally Baire if there is a class tree S such that S, T project to complementary subsets of reals in all set generic extensions. A set A of reals is universally Baire if it is a projection of a universally Baire tree.

This definition is not stated within ZFC proper. The results below have to be carefully restated to make the intended sense in ZFC. This is a standard procedure. The basic properties of the universally Baire sets are stated in the following

Fact C.3. (ZFC+LC) Projective sets are universally Baire. Moreover

• The universally Baire sets are closed under complements, countable intersections and unions and existential and transfinite real game quantifiers.

• Every universally Baire relation can be uniformized by a universally Baire function, and every transfinite real game of a fixed countable length with universally Baire payoff set has a universally Baire winning strategy for one of the players.

The statements in the Fact follow mostly immediately from C.8 and C.9. For example suppose that B ⊂ ℝ × ℝ^{α} is a universally Baire set, where α is a countable ordinal. The set A = {r ∈ ℝ : Adam has a winning strategy in the transfinite real game of length α given by the section B_{r}} is evaluated correctly in every model of the Axiom of Real Determinacy containing all the reals, since such a model must have a winning strategy for one of the sides in the game by C.9 and what it thinks is a winning strategy really is such because the model contains all the possible counterplays. By C.8 then, the set A is universally Baire.

One trick involving universally Baire sets is repeated at many places in this paper. Suppose A = p[T] = ℝ \ p[S] is a universally Baire set and M is an elementary submodel of some large enough structure. Then all generic extensions of the model M evaluate the membership in the set A correctly. That is, if r ∈ M[G] is a real in such a generic extension, then by the universal Baireness applied in the model M, either M[G] ⊧ r ∈ p[T] and then clearly r ∈ A, or M[G] ⊧ r ∈ p[S] and then r ∉ A.

C.1 The absoluteness results

Fact C.4. [35] Suppose that there is a proper class of Woodin cardinals. Let T_{0}, T_{1}, … T_{n−1} be universally Baire trees. The theory of the model ⟨L(ℝ)[p[T_{i}] : i ∈ n], ∈, p[T_{i}] : i ∈ n⟩ with arbitrary real parameters is the same in all set generic extensions.

One application of this fact is that given a universally Baire set A ⊂ ℝ and a forcing P, there is a P -name Ȧ for a set of reals which is the projection of the same ground model universally Baire tree as the set A, and this name Ȧ does not depend on the particular choice of the tree.

Fact C.5. [6] Suppose that there is a proper class of Woodin cardinals. Let T be a universally Baire tree. For every set generic extension V [G] of V via a proper notion of forcing, there is an elementary embedding j : (L(ℝ)[p[T]])^{V} → (L(ℝ)[p[T]])^{V [G]} which fixes the ordinals.

Thus, compared to the previous fact, if the forcing extension is effected via proper forcing, the theory of the model L(ℝ)[p[T]] stays the same even if ordinal parameters are allowed. It is clear that the elementary embedding j is unique.

Fact C.6. [34] [38] Suppose that there is a proper class of measurable Woodin cardinals. Suppose that ϕ is a Σ_{1}^{2} sentence with a possible real parameter and a possible universally Baire predicate. If ϕ holds in some generic extension then it holds in every set generic extension satisfying the Continuum Hypothesis.

C.2 The determinacy results

Fact C.7. [35] Suppose that there are infinitely many Woodin cardinals with a measurable cardinal above. Suppose T is a universally Baire tree. Then L(ℝ)[p[T]] satisfies the Axiom of Determinacy.

Fact C.8. [34] Suppose that there is an inaccessible cardinal δ which is a limit of Woodin cardinals and < δ-strong cardinals. Suppose that T is a universally Baire tree. Then there is a pointclass Γ including p[T] and consisting solely of universally Baire sets such that Γ = P(ℝ) ∩ L(ℝ)(Γ) and L(ℝ)(Γ) satisfies the Axiom of Real Determinacy and DC.

The Axiom of Real Determinacy states that all infinite two player games whose moves are real numbers, are determined. This axiom is implication-wise and consistency-wise strictly stronger than the Axiom of Determinacy. The consequence of this axiom most important for this paper is

Fact C.9. [34] (ZF+DC) The Axiom of Real Determinacy is equivalent to each of the following statements:

• Axiom of Determinacy and every set in the plane can be uniformized

• for every countable ordinal, all two player integer games whose length is that ordinal are determined

• Axiom of Determinacy and every set of reals is Souslin.

Here a set is Souslin if it is a projection of a tree on some (κ × ω)^{<ω}, where κ is an ordinal.

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