Duality and the pcf theory

Saharon Shelah and Jindřich Zapletal ^{∗}

March 17, 2002

Abstract

We consider natural cardinal invariants 𝔥𝔪_{n} and prove several duality theorems, saying roughly: if I is a suitably definable ideal and provably cov(I) ≥ 𝔥𝔪_{n}, then non(I) is provably small. The proofs integrate the determinacy theory, forcing and pcf theory.

1 Introduction

[6] considered the following cardinal invariant of the continuum. Let c : [2^{ω}]^{2} → 2 be the partition defined by c(x, y) = Δ(x, y) mod 2 where Δ(x, y) = min{m ∈ ω : x(m) ≠ y(m)}, and let J be the σ-ideal σ-generated by the homogeneous sets. It is not difficult to see that J is a proper σ-ideal; in fact c is the simplest continuous partition of [2^{ω}]^{2} with this property. [6] defined the cardinal invariant 𝔥𝔪 as the covering number of the ideal J. The invariant 𝔥𝔪 occurs naturaly in several contexts: it is the minimal number of Lipschitz-with-constant-1 functions from 2^{ω} to itself such that their graphs and graphs of their inverses cover the whole square 2^{ω} ×2^{ω}. It is also the minimal number of convex subsets necessary to cover a certain closed subset of the Euclidean plane. [6]

It is consistent with ZFC that 𝔥𝔪 < 𝔠; not surprisingly [7], this is exactly what happens in the Sacks model [6]. On the other hand, 𝔥𝔪 is a very large cardinal invariant in that provably 𝔥𝔪^{+} ≥ 𝔠. Here we prove a duality theorem similar to the one from [7].

Theorem 1.1. Suppose that I is a projectively generated projectively defined ideal such that ZFC+LC proves that cov(I) ≥ 𝔥𝔪. Then ZFC+LC proves that non(I) ≤ ℵ_{3}.

The pattern persists to some variations of 𝔥𝔪. For every natural number n > 0 let c_{n} : [2^{ω}]^{2} → n be the function defined by c_{n}(x, y) = Δ(x, y) mod n and let J_{n} be the σ-ideal on 2^{ω} σ-generated by the sets X such that c′′_{n}X ≠ n. Define J_{1} to be the ideal of countable sets. Define 𝔥𝔪_{n} to be the covering number of the ideal J_{n}. It is not difficult to show that 𝔠 = 𝔥𝔪_{1} ≥ 𝔥𝔪 = 𝔥𝔪_{2} ≥ 𝔥𝔪_{3} ≥ 𝔥𝔪_{4} ≥ … and (𝔥𝔪_{n})^{n−1} ≥ 𝔠 for every number n. Again, 𝔥𝔪_{n} can be rewritten as the smallest number of functions in a certain class necessary to cover the cube (2^{ω})^{n} with their graphs. We have

∗The first author is supported by The Israel Science Foundation founded by the Israel Academy of Sciences and Humanities. Publication 791. The second author is partially supported by grants GA ČR 201-00-1466, NSF DMS-0071437 and a CLAS UF research award.

Theorem 1.2. Let n > 0. Suppose that I is a projectively generated projectively defined ideal such that ZFC+LC proves that cov(I) ≥ 𝔥𝔪_{n}. Then ZFC+LC proves that non(I) ≤ ℵ_{n+1}.

In case of n = 1 this improves the original duality theorem of [7], and the obtained bound ℵ_{2} is optimal. Finally,

Theorem 1.3. Suppose that I is a projectively generated projectively defined ideal such that ZFC+LC proves that cov(I) ≥ min_{n} 𝔥𝔪_{n}. Then ZFC+LC proves that non(I) ≤ ℵ_{ω_{2}+1}.

The proofs of the above theorems integrate the determinacy theory, forcing, and pcf theory.

The notation in the paper follows the set theoretic standard of [1]. Whenever X is a set, x ∈ X its element and f : (2^{ω})^{X\{x}} → 2^{ω} is a function, by the graph of the function f we mean the collection {r⃗ ∈ (2^{ω})^{X} : r⃗(x) = r⃗ ↾ X \ {x}}. A σ-ideal I on 2^{ω} is called homogeneous if for every I-positive Borel set B there is a function f : 2^{ω} → B such that the preimages of I-small sets are I-small [7]. For a σ-ideal on the reals, the cardinal cov(I) and non(I) denote the smallest size of a family of sets in the ideal covering the whole real line, and the smallest size of an I-positive set respectively. LC denotes the use of a sufficiently strong large cardinal axiom; a supercompact cardinal is always strong enough.

2 The forcings associated with 𝔥𝔪_{n}

The following is a basic simple observation.

Lemma 2.1. Fix a number n > 0.

1. The ideal J_{n} is σ-generated by closed sets.

2. The ideal J_{n} is proper.

Proof. The partitions c_{n} are continuous. Therefore, if X ⊂ 2^{ω} is a set in the ideal J_{n} (the image c′′_{n}[X]^{2} does not contain some number m ∈ n), then even its closure is in the ideal J_{n} (its image under c_{n} still does not contain that number m). The first item immediately follows.

For the second item, consider the bijection g_{n} between (2^{ω})^{n} and 2^{ω} defined by g_{n}(x_{0}, x_{1}, … x_{n−1}) = y where y(m) = x_{l}(k) where k is the integer part of m/n and l = m mod n. What is the preimage of the ideal J_{n}? Let X ⊂ 2^{ω} be a closed set such that c′′_{n}X misses at least one value, say k ∈ n. It is not difficult to see that the preimage of the set X under the bijection g is a graph of a partial Borel function from (2^{ω})^{n\{k}} to 2^{ω}: if x⃗ = y⃗ are two n-tuples such that g_{n}(x⃗), g_{n}(y⃗) ∈ X and x⃗(m) = y⃗(m) for all numbers m ∈ n different from k, then necessarily x⃗(k) = y⃗(k) as well, since otherwise c_{n}(g_{n}(x⃗), g_{n}(y⃗)) = k, contradicting the choice of the set X.

We will now show that the product of n copies of the Cohen forcing forces the image of the generic n-tuple r⃗_{gen} under the bijection g_{n} to fall out of all ground model coded J_{n}-small sets: if X is such a closed set, then g_{n}^{−1}X ⊂ (2^{ω})^{n} is a graph of a partial Borel function, and by the mutual genericity of the reals on the n-tuple r⃗_{gen} the n-tuple does not belong to this graph. Obviously, we could have used any other forcing adding a real in the place of Cohen forcing.

The second item now immediately follows. Let {X_{m} : m ∈ ω} be a countable collection of closed sets in the ideal J_{n}. The previous paragraph shows that the product of n copies of the Cohen forcing forces that 2^{ω} ≠ ⋃_{n} Ẋ_{n}. By the analytic absoluteness, 2^{ω} ≠ ⋃_{n} X_{n} as desired.

Thus by the results of [8], the partial order P_{n} of J_{n}-positive Borel sets ordered by inclusion adds a single real which falls out of all J_{n}-small sets, it is proper, and in the case that the ideal J_{n} is homogeneous, the countable support iteration of poset P_{n} is an optimal way to increase the invariant 𝔥𝔪_{n}. The invariants 𝔥𝔪_{n} are quite special because the ideal J_{n} does turn out to be homogeneous:

Lemma 2.2. (Geschke) Suppose d : [2^{ω}]^{2} → n is a continuous partition and A ⊂ 2^{ω} is an analytic set which is not covered by countably many sets X for which d′′X ≠ n. Then there is a continuous injection i : 2^{ω} → A such that for all sequences x ≠ y in 2^{ω}, c_{n}(x, y) = d(i(x), i(y)).

Proof. Let T ⊂ (2 × ω)^{<ω} be a tree whose projection is the set A. Pruning the tree if necessary, we may assume that for all nodes t ∈ T the projection of the tree T ↾ t is not covered by countably many sets X for which d′′X ≠ n. Now for every sequence u ∈ 2^{<ω} construct sequences f(u) ∈ 2^{<ω} and g(u) ∈ ω^{<ω} such that

1. the sequences f(u) and g(u) have the same length and ⟨f(u), g(u)⟩ ∈ T

2. u_{0} ⊂ u_{1} implies f(u_{0}) ⊂ f(u_{1}) and g(u_{0}) ⊂ g(u_{1})

3. if the length of the sequence u is modulo n equal to some m ∈ n, then f(u^{⌢}0) and f(u^{⌢}1) are inclusion-incomparable sequences such that for all x, y ∈ 2^{ω} extending f(u^{⌢}0) and f(u^{⌢}1) respectively we have d(x, y) = m.

The construction is easy and proceeds by tree induction along 2^{<ω} on the sequence u. Let f(0) = g(0) = 0. Suppose that f(u), g(u) have been constructed and the length of u is modulo n equal to some m ∈ n. To find the sequences f(u^{⌢}0), f(u^{⌢}1) and g(u^{⌢}0), g(u^{⌢}1), first use the properties of the tree T to find two branches ⟨x, a⟩ and ⟨y, b⟩ of the tree through the node ⟨f(u), g(u)⟩ ∈ T such that x ≠ y and d(x, y) = m. By the continuity of the partition d there must be initial segments f(u^{⌢}0) and f(u^{⌢}1) of x and y respectively satisfying the third item above. Let g(u^{⌢}0) and g(u^{⌢}1) be the respective initial segments of a and b.

Now clearly the function i : 2^{ω} → A defined by i(x) = ⋃_{n∈ω} f(x ↾ n) is the desired injection.

Corollary 2.3. The ideal J_{n} is homogeneous. The forcing P_{n} is homogeneous.

Proof. Suppose that B ⊂ 2^{ω} is a J_{n}-positive Borel set, and let i : 2^{ω} → B be a continuous injection respecting the partition c_{n}; such an injection exists by the previous lemma. Clearly, the preimages of J_{n}-small sets must be J_{n}-small, and therefore the ideal J_{n} is homogeneous. The function π defined by π(C) = π′′C for every J_{n}-positive Borel set C ⊂ 2^{ω}, is an isomorphism between the poset P_{n} and P_{n} below the condition π(2^{ω}) ⊂ B. Note that Borel injective images of Borel sets are Borel.

The following is now a consequence of the previous lemmas, Theorem of [5] and Theorem 7.4 of [7].

Corollary 2.4. (ZFC+LC) Suppose n > 0 and 𝔵 is a tame invariant. If 𝔵 < 𝔥𝔪_{n} holds some forcing extension, then it holds in the countable support iterated P_{n}-extension.

As an aside, let us define the cardinal invariant 𝔥𝔪_{ω} in the following way. Let {a_{n} : n ∈ ω} be an arbitrary partition of ω into infinite sets, and for distinct sequences x, y ∈ 2^{ω} define c_{ω}(x, y) = n if the smallest number m such that x(m) ≠ y(m) belongs to the set a_{n}. Let J_{ω} be the σ-ideal generated by the sets X ⊂ 2^{ω} such that c′′_{ω}X ≠ ω, and define 𝔥𝔪_{ω} to be the covering number of this ideal. The previous results all apply to the case of the invariant 𝔥𝔪_{ω}, and they also show that 𝔥𝔪_{ω} is independent of the particular choice of the initial partition {a_{n} : n ∈ ω}. However, it is impossible to generalize the theorems stated in the introduction to the case of 𝔥𝔪_{ω}, since the results of the next section will not be applicable.

3 The iterated Fubini powers of the ideals J_{n}

The theorems stated in the introduction are proved using a careful analysis of the countable support iteration of the forcings P_{n}. We will need to find an upper bound, in terms of the ℵ function, of the uniformities of the iterated Fubini powers of the ideals J_{n}. But in fact, in order to streamline the notation, we will not deal with the ideals J_{n} directly. For every n > 0 let K_{n} be the σideal on (2^{ω})^{n} generated by the graphs of Borel partial functions from (2^{ω})^{n\{k}} to 2^{ω} for all k ∈ n. By the argument in the proof of Lemma 2.1, g^{−1}J_{n} ⊂ K_{n} and certainly non(J_{n}) ≤ non(K_{n}) and for each countable ordinal α, non(J_{n}^{α}) ≤ non(K_{n}^{α}). Thus it will be enough to find the uniformities of the ideals K_{n}^{α}. Recall:

Definition 3.1. Suppose that I is a σ-ideal on the real line and α ∈ ω_{1} is an ordinal. The α-th iterated Fubini power of the ideal I is the σ-ideal I^{α} on ℝ^{α} consisting of those sets A ⊂ ℝ^{α} for which Adam has a winning strategy in the two person game G(A) of length α. In β-th round of the game G(A) first Adam plays a set in the ideal I and then Eve chooses a real not in the set. Eve wins if the sequence of her answers belongs to the set A.

Lemma 3.2. Let n > 0 be a natural number and α be a countable ordinal. Then non(K_{n}^{α}) ≤ ℵ_{n+1}.

Proof. The following is the key fact in the argument:

Fact 3.3. (Shelah) Let κ > ω_{1} be a regular cardinal. For every countable limit ordinal α ∈ ω_{1} there is a sequence C⃗ = ⟨C_{δ} : δ ∈ κ⟩ such that

1. C_{δ} ⊂ δ is a closed set of ordertype ≤ α

2. for every ordinal δ ∈ κ and every accumulation point γ ∈ C_{δ}, the set C_{γ} is just C_{δ} ∩ γ

3. for every closed unbounded set E ⊂ κ there is an ordinal δ ∈ E such that the set C_{δ} is cofinal in δ of ordertype α and it is a subset of E.

This Fact has been announced in [3], page 136, remark 2.14A. The proof is in [2], available from the Mathematics ArXiv. Now fix a number n > 0 and a countable ordinal α ∈ ω_{1}, without loss assuming that α is limit. For every number m ∈ n use the Fact to choose a club guessing sequence C⃗_{m} = ⟨C_{δ}^{m} : δ ∈ ω_{m+2}⟩ on ω_{m+2}. We may certainly assume that 𝔠 ≥ ℵ_{n+1} and so we can choose a sequence s⃗ = ⟨s_{δ} : δ ∈ ω_{n+1}⟩ of pairwise distinct reals–elements of 2^{ω}. For every n-tuple δ⃗ ∈ ∏_{m∈n} ω_{m+2} such that the sets C_{δ⃗(m)}^{m} have ordertype α,

let ⟨δ⃗(m)(β) : β ∈ α⟩ enumerate the nonaccumulation points of these sets in the increasing order, and let r⃗δ⃗ ∈ (((2^{ω})^{n})^{α}) be the α-sequence whose β-th element is the point ⟨s_{δ⃗(m)(β)} : m ∈ n⟩ in the space (2^{ω})^{n}. It will be enough to show

that the set {r⃗δ⃗ : δ⃗ ∈ ∏_{m∈n} ω_{m+2}} is J_{n}^{α}-positive.

To prove this, for every Adam’s strategy σ in the transfinite game we need to find an n-tuple δ⃗ such that the sequence r⃗δ⃗ is a legal counterplay against the strategy σ. So fix the strategy σ and by downward induction on m ∈ n find

1. a continuous increasing ∈-tower M⃗_{m} = ⟨M_{δ}^{m} : δ ∈ ω_{m+2}⟩ of elementary submodels of a large enough structure, each of them of size ℵ_{m+1} and such that M_{δ}^{m} ∩ ω_{m+2} ∈ ω_{m+2}. In particular, we require σ ∈ M_{0}^{m}. Let E_{m} = {δ ∈ ω_{m+2} : M_{δ}^{m} ∩ ω_{m+2} = δ}; this is a closed unbounded subset of ω_{m+2}.

2. an ordinal δ_{m} ∈ ω_{m+2} such that the set C_{δ_{m}}^{m} ⊂ δ_{m} is cofinal of ordertype α and it is a subset of the club E_{m}. For every two numbers k ∈ m ∈ n we demand that δ_{m} ∈ M_{0}^{k}.

Let δ⃗ = ⟨δ_{m} : m ∈ n⟩. We claim that the sequence r⃗δ⃗ is the desired legal counteplay against the strategy σ. So look at an arbitrary round β ∈ α and suppose that the sequence does constitute a legal counterplay up to this point. What happens at round β? The important observation is that for every integer k ∈ n, the play up to this round is in the model M_{δ⃗(k)(β)}^{k} since it is defined from objects that belong to this model. In particular, one of the parameters in the definition is the set C_{δ⃗(k)}^{k} ∩ δ⃗(k)(β) which is in the model by the coherence requirement (2) in Fact 3.3. The strategy σ now commands Adam to play partial Borel functions {f_{mk} : m ∈ n, k ∈ ω} such that f_{mk} is a function from (2^{ω})^{n\{m}} to 2^{ω}. We must show that the point r⃗δ⃗(β) is not contained in the graph of any of these functions. So choose integers m ∈ ω and k ∈ n. Consider the set Y = {f_{mk}(u⃗, v⃗) : u⃗ ∈ (2^{ω})^{k} is a sequence all of whose entries are on the s⃗-sequence, indexed by ordinals smaller than ω_{k+1} and v⃗ ∈ (2^{ω})^{n\(k+1)} is a sequence all of whose entries are on the s⃗-sequence, indexed by ordinals in the set ⋃_{k+2∈l∈n+2} C_{δ⃗(l)}^{l}}. This set is of size < ω_{k+2} and it belongs to the model

M_{δ⃗(k)(β)}^{k}. Thus Y ⊂ M_{δ⃗(k)(β)}^{k}, in particular s_{δ⃗(k)(β)} ∉ Y, which by the definition of the set Y means that the point r⃗δ⃗(β) = ⟨s_{δ⃗(l)(β)} : l ∈ n⟩ is not on the graph of the function f_{mk} as desired. The lemma follows.

Corollary 3.4. Let n > 0 be a natural number and let α be a countable ordinal. non(J_{n}^{α}) ≤ ℵ_{n+1}.

In order to analyse the countable support iteration in which the forcings P_{n} alternate, we need to change the approach a little bit:

Definition 3.5. Suppose that I⃗ is an ω-sequence of σ-ideals on the reals and α is a countable ordinal. The ideal I⃗^{α} on ℝ^{ω·α} consists of those sets A ⊂ ℝ^{ω·α} for which Adam has a winning strategy in the transfinite two person game G(A) of length ω · α. In the ω · β + n-th round of this game, Adam plays a set in the ideal I⃗(n) and Eve responds with a real which is not in this set. Eve wins if the sequence of her answers belongs to the set A.

Let K⃗ be the ω-sequence of ideals given by K⃗(n) = J_{n+1}.

Lemma 3.6. Let α be a countable ordinal. Then non(K⃗^{α}) ≤ ℵ_{ω_{2}+1}.

Proof. Let α be an arbitrary countable ordinal. First we need to fix several objects whose existence is provable in ZFC.

1. An increasing sequence ⟨κ_{δ} : δ ∈ ω_{2}⟩ of regular cardinals below ℵ_{ω_{2}} such that the true cofinality of their product modulo the bounded ideal on ω_{2} is ω_{ω_{2}+1}, from [4], Chapter II, Theorem 1.5. Fix a cofinal increasing sequence ⟨h_{γ} : γ ∈ ω_{ω_{2}+1}⟩ of functions in ∏_{δ} κ_{δ} witnessing this fact.

2. A club guessing sequence C⃗ = ⟨C_{γ} : γ ∈ ω_{ω_{2}+1}⟩ from Fact 3.3. The sequence will guess closed unbounded subsets of ω_{ω_{2}+1} by segments of length ω · ω · α. For every ordinal γ ∈ ω_{ω_{2}+1} let γ(β) denote the β-th nonaccumulation point of the set C_{γ}.

3. A club guessing sequence D⃗ = ⟨D_{δ} : δ ∈ ω_{2}⟩ from Fact 3.3. The sequence will guess closed unbounded subsets of ω_{2} by segments of length ω · ω · α again, with similar notational convention as in the previous item.

4. Without harm we may assume that 𝔠 > ℵ_{ω_{2}}. So let us fix a sequence s⃗ = ⟨s_{γ} : γ ∈ ω_{ω_{2}+1}⟩ of pairwise distinct reals (elements of 2^{ω}).

Now suppose that γ ∈ ω_{ω_{2}+1} and δ ∈ ω_{2} are ordinals such that the ordertypes of the sets C_{γ} and D_{δ} are both ω ·ω ·α. Define a ω ·α sequence r⃗_{γδ} by setting its ω · β + n-th element to be the point in the space (2^{ω})^{n+1} whose k-th coordinate for every number k ∈ n + 1 is the real number on the s sequence indexed by the ordinal h_{γ(ω·(ω·β+n)+k)}(δ(ω · (ω · β + n) + n − k)). We will show that the collection {r⃗_{γδ} : γ ∈ ω_{ω_{2}+1}, δ ∈ ω_{2}} is (J⃗)^{α}-positive, proving the lemma. This means that for every Adam’s strategy σ in the transfinite game we must find ordinals γ ∈ ω_{ω_{2}+1} and δ ∈ ω_{2} such that the sequence r⃗_{γδ} is a legal counterplay against the strategy.

Fix a continuous increasing ∈-tower ⟨M_{γ} : γ ∈ ω_{ω_{2}+1}⟩ of elementary submodels of large enough structure, each of them of size ℵ_{ω_{2}} and such that M_{γ}∩ω_{ω_{2}+1} ∈ ω_{ω_{2}+1}. In particular, σ ∈ M_{0}. Let E = {γ ∈ ω_{ω_{2}+1} : M_{γ} ∩ ω_{ω_{2}+1} ∈ ω_{ω_{2}+1}}. Since this is a closed unbounded subset of ω_{ω_{2}+1}, there must be an ordinal γ such that the set C_{γ} ⊂ γ is cofinal of ordertype ω · ω · α and C_{γ} ⊂ E.

Also, fix a continuous increasing ∈-tower ⟨N_{δ} : δ ∈ ω_{2}⟩ of elementary submodels of large enough structure, each of them of size ℵ_{1} and such that N_{δ} ∩ ω_{2} ∈ ω_{2}. In particular, σ, γ ∈ N_{0}. Let F = {δ ∈ ω_{2} : N_{δ} ∩ ω = δ}. Since this is a closed unbounded set, there must be an ordinal δ ∈ ω_{2} such that the set D_{δ} ⊂ δ is cofinal of ordertype ω · ω · α and D_{δ} ⊂ F.

We claim that the sequence r⃗_{γδ} is a legal counterplay against the strategy σ. Well, consider the situation at round ω · β + n for some ordinal β ∈ α and number n ∈ ω. Suppose that up to this point, the sequence constituted a legal partial play; we want to see that it will provide a legal answer even in this round. The strategy σ commands Adam to play partial Borel functions {f_{mk} : k ∈ n + 1, m ∈ ω} such that for each k ∈ n + 1 and each m ∈ ω the function maps (2^{ω})^{n+1\{k}} to (2^{ω}). We must show that the n + 1-tuple t⃗ = r⃗_{γδ}(ω · β + n) ∈ (2^{ω})^{n+1} is not on the graph of any of these functions, that is t⃗(k) ≠ f_{mk}(t⃗ ↾ (n + 1 \ {k})).

To this end, fix integers k ∈ n + 1 and m ∈ ω. Define a function g ∈ ∏_{δ} κ_{δ} by letting g(δ) to be the supremum of the set {ξ ∈ κ_{δ} : s_{ξ} = f_{mk}(u⃗, v⃗), where u⃗ ∈ (2^{ω})^{k} is a sequence all of whose entries are on the s-sequence, and are indexed by ordinals < sup{κ_{δ′} : δ′ ∈ δ}, and v⃗ ∈ (2^{ω})^{(n+1)\(k+1)} is a sequence all of whose entries are on the s-sequence and are indexed in the ordinals in the range of the functions {h_{γ′} : γ′ ∈ C_{γ} ∩ γ(ω · (ω · β + n) + k)}}. It is immediate that this set has size < κ_{δ} and so the function g is well-defined. There are two important points.

1. g ∈ M_{γ(ω·(ω·β+n)+k)}. This so happens because the function is defined from objects contained in the model, in particular from the set C_{γ} ∩γ(ω·(ω·β + n) + k) which belongs to the model by the coherence of the C-sequence.

2. g ∈ N_{δ(ω·(ω·β+n)+k)} by the same reason as in the previous item, this time using the coherence of the D-sequence.

By the first point, the function g ∈ ∏_{δ} κ_{δ} is dominated by the function h_{γ(ω·(ω·β+n)+k)} from some ordinal on. By the second point, this ordinal must be smaller than δ(ω·β+n)+k). By the definition of the function g and the sequence t⃗ = r⃗_{γδ}(ω · β + n) then, it must be the case that t⃗(k) ≠ f_{mk}(t⃗ ↾ (n + 1 \ {k})) as desired. The lemma follows.

Let J⃗ be the ω-sequence of ideals given by J⃗(n) = J_{n+1}.

Corollary 3.7. Let α be a countable ordinal. non(J⃗^{α}) ≤ ℵ_{ω_{2}} + 1.

Let K_{ω} be the σ-ideal on (2^{ω})^{ω} σ-generated by the graphs of partial Borel functions from (2^{ω})^{ω\{n}} to 2^{ω}, for all numbers n ∈ ω. The following lemma is an observation complementary to the previous results in this section. It shows that the ideal K_{ω} is different from the ideals K_{n} in that its uniformity can be arbitrarily large:

Lemma 3.8. 𝔭 ≤ non(K_{ω}).

Proof. Suppose that A ⊂ (2^{ω})^{ω} is a set of size < 𝔭. We must produce partial Borel functions {f_{n} : n ∈ ω} such that f_{n} : (2^{ω})^{ω\{n}} → 2^{ω} and every point in the set A is on the graph of one of them. First, use the Axiom of Choice to find a set B ⊂ A so that

1. for all sequences x⃗, y⃗ ∈ B if x⃗(m) = y⃗(m) for all but finitely many integers m then x⃗ = y⃗

2. B ⊂ A is a maximal set with the previous property.

For each number n let g_{n} be a partial function from (2^{ω})^{ω\{n}} to 2^{ω} defined by g_{n}(x⃗) = z if the sequence y⃗_{0} = x⃗ ∪ {⟨n, z⟩} is in the set A and for some sequence y⃗_{1} in the set B, y⃗_{0}(m) = y⃗_{1}(m) for all numbers m ≥ n. By the property (1) of the set B, this formula does correctly define the functions g_{n}, and by the property (2), every point in the set A is on the graph of all but finitely many of these functions. A standard application of Martin’s Axiom for σ-centered partial orders–see for example [7], Claim B.13–now gives Borel functions f_{n} such that g_{n} ⊂ f_{n} for all numbers n ∈ ω. The lemma follows.

4 The duality theorems

The proof of Theorem 1.2 follows the argument for the first duality theorem in the Applications section of [7]. Suppose that n > 0 is a natural number and I is a projectively generated projectively defined ideal on the reals such that ZFC+LC proves cov(I) ≥ 𝔥𝔪_{n}. We want to argue in ZFC+LC that non(I) ≤ ℵ_{n+1}. For this, it is necessary to analyse the countable support iteration of the P_{n} forcing. The following Facts use the terminology and arguments from [7].

Fact 4.1. (ZFC+LC) For every countable ordinal α ∈ ω_{1} and every projective set A ⊂ (2^{ω})^{α}, either the set A contains a Borel J_{n}-perfect subset, or it belongs to the ideal J_{n}^{α}.

This holds simply because the ideal J_{n} is generated by closed sets.

Fact 4.2. (ZFC+LC) For every countable ordinal α ∈ ω_{1} the ideal J_{n}^{α} is homogeneous.

This happens because the ideal J_{n} is homogeneous. It means in particular that for every Borel J_{n}-perfect set B ⊂^{(} 2^{ω})^{α} the invariant non(J_{n}^{α}) is equal to non(J_{n}^{α}), the smallest size of a J_{n}^{α}-positive subset of B.

Fact 4.3. (ZFC+LC) There is a countable ordinal α ∈ ω_{1}, a Borel J_{n}-perfect subset B ⊂ (2^{ω})^{α} and a Borel function f : B → ℝ such that f-preimages of I-small sets are J_{n}^{α}-small.

This is true because after the countable support iteration of the P_{n}-forcing of length 𝔠^{+}, it is still the case that cov(I) ≥ 𝔥𝔪_{n} and so there must be a name for a real which falls out of all ground model coded I-small sets. This name must be represented by a Borel set and a function as in the above Fact.

Now fix α, B and f from the previous Fact. By Fact 2.3 and Corollary 3.4, there is a J_{n}^{α}-positive subset A ⊂ B of size ≤ ℵ_{n+1}. By the properties of the function f, the set f′′A must be an I-positive set of size ≤ ℵ_{n+1}, and Theorem 1.2 follows.

The argument for Theorem 1.3 is similar, using the analysis of the countable support iteration in which the forcings P_{n} alternate and Corollary 3.7. Theorem 1.1 is just a special case of Theorem 1.2.

References

[1] Thomas Jech. Set Theory. Academic Press, San Diego, 1978.

[2] Saharon Shelah. Analytical Guide and updates for ”Cardinal Arithmetic”. Shelah [Sh:E12].

[3] Saharon Shelah. Non-structure Theory, volume accepted. Oxford University Press.

[4] Saharon Shelah. Cardinal Arithmetic. Clarendon Press, Oxford, 1994.

[5] Slawomir Solecki. Covering analytic sets by families of closed sets. Journal of Symbolic Logic, 59:1022–1031, 1994.

[6] Stefan Geschke, Menachem Kojman, Wieslaw Kubiś, and R. Schipperus. Convex decompositions in the plane and continuous pair colorings of the irrationals. preprint, 2001.

[7] Jindřich Zapletal. Determinacy and cardinal invariants. 2002.

[8] Jindřich Zapletal. Forcing with ideals of closed sets. Commentationes Mathematicae Universitatis Carolinae, 2002.