## Oberseminar mathematische Logik

### Organizers

- Prof. Dr. Peter Koepke
- Dr. Philipp Lücke
- Dr. Philipp Schlicht

### Time and location

Mondays 16.30-18.00 in room 0.008, Endenicher Allee 60.

The participants of the seminar are welcome for coffee and tea in the Plückerraum 1.012 at 16.00 before the talks.

### Talks

- 17 October: no seminar
- 24 October: Peter Holy (Bonn) - How to and how not to turn proper classes into sets

We investigate the question whether it is possible to start with a model of set theory and perform class forcing over this model so that a proper class of the ground model becomes a set in the generic extension.

It is easily seen that this is not possible over models of ZFC, but perhaps somewhat surprisingly, we show that this is possible in the absence of the power set axiom, i.e. over models of ZFC^{-}. We show that we can perform class forcing over H(ω_{1}) so that the ground model reals become a set in any generic extension. However the notion of class forcing used does not satisfy the forcing theorem.

Furthermore, we provide several sufficient conditions that preclude the above from happening also in the absence of the power set axiom, ensuring that no proper class of the ground model becomes a set through class forcing. We do not know whether the forcing theorem is such a sufficient condition as well.

This is joint work with Philipp Schlicht. - 31 October: Philipp Lücke (Bonn) - Simple formulas defining complicated sets

We consider the question whether large cardinal axioms imply that certain complicated sets cannot be defined by simple formulas. More precisely, given an uncountable regular cardinal κ, we study the collection of subsets of H(κ^{+}) that are definable by Σ_{1}-formulas with parameters in H(κ)∪{κ} in the presence of large cardinals. In this talk, I will present results showing that (for certain cardinals κ) large cardinal assumptions imply that these classes do not contain well-orders of the reals, reductions of turbulent group actions to the isomorphism relation between models of size κ or homeomorphisms between the κ-Baire space^{κ}κ and the κ-Cantor space^{κ}2.

This is partly joint work in progress with Luca Motto Ros (Turin) and partly joint work with Ralf Schindler (Münster) and Philipp Schlicht (Bonn). - 07 November: no seminar
- 14 November: no seminar
- 21 November: no seminar
- 28 November: Philipp Schlicht (Bonn) - Large cardinals defined by chains of models

We study large cardinals defined by the existence of certain chains of models, in particular chains of models defined by iterated filter properties. We show that these large cardinals form a hierarchy between weakly compact and measurable cardinals and compare them with the Ramsey-like cardinals studied by Victoria Gitman.

This is joint work with Peter Holy. - 05 December: Hazel Brickhill (Bristol) - Hyper-stationary sets and Generalised Square Sequences in the Constructible Universe

I will introduce generalised stationary and closed unbounded sets and some of their basic properties. We will then look at where these sets appear in L. To do this we will define a square sequence using hyper-clubs that generalises Jensen's square below a non-weakly compact, used to show that in L a cardinal is stationary reflecting iff it is weakly compact. I will then show how, in L, we can construct such generalised square sequences, which witness that a cardinal which is not Π^{1}_{n}-indescribable does not reflect n-stationary sets. Unlike Jensen's square, our constructions here are not fine-structural. I will go on to explain how this argument can be generalised to γ-stationary reflection, with an appropriate definition for Π^{1}_{n}-indescribability. - 12 December: Alexander Block (Hamburg) - The bimodal logic of forcing and grounds

The (ZFC provable) modal logic of forcing is defined as the collection of all basic modal formulas A such that every sentence arising from substituting every propositional variable occurring in A by a set theoretic sentence and interpreting the modal box operator as "in every forcing extension [...] holds" is provable from ZFC. Analogously we can define the modal logic of grounds. The modal logic of forcing was proved to be equal to the modal logic S4.2 by Hamkins and Löwe, the modal logic of grounds is known to be equal to S4.2 by recent work by Usuba.

We can investigate a combined logic by considering the bimodal logic of forcing and grounds formulated in a multimodal language with two distinct modal box operators, one of which shall be interpreted in the sense of forcing and the other one in the sense of grounds. Although we know the monomodal fragments of this logic, it is not yet fully known which modal axioms govern the interaction of its two modalities. In my talk I will present the known constraints for such interaction principles, in particular an upper bound for the bimodal logic of forcing and grounds. - 19 December: Regula Krapf (Bonn) - Class forcing over models of second-order arithmetic and preservation of the perfect set property

In this talk we provide a framework for class forcing over models of second-order arithmetic. Since second-order arithmetic enhanced by the boldface Π^{1}_{1}-perfect set property is equiconsistent with ZFC, we are interested in the preservation of this property under class forcing. Following results from Castiblanco/Schlicht we show that many arboreal forcing notions preserve the perfect set property. On the other hand, using reshaping and almost disjoint coding forcing the perfect set property can be destroyed. - 09 January: Ralf Schindler (Münster) - A Hamel basis for the reals without choice

The Cohen-Halpern-Levy model N has an infinite set of reals without a countable subset. Answering a question of D. Pincus and K. Prikry from 1975, we show that there is a Hamel basis in N. This is joint work with Liuzhen Wu and Liang Yu, inspired by earlier joint work with Mariam Beriashvili. DC fails in N, and it remains open if in the base theory ZF+DC, the existence of a Hamel basis implies that the reals can be wellordered. - 13 January: Joel Hamkins (New York) - The downward directed grounds hypothesis
- 16 January: no seminar
- 23 January: Philipp Lücke (Bonn) - Measurable cardinals and good wellorders

We study the interplay between large cardinals and the existence of very simply definable wellorderings of power sets of cardinals. In this talk, I will present a result that isolates all restrictions on the existence of so-called good Σ_{1}-wellorders that are provably implied by the existence of a measurable cardinal.

This is joint work in progress with Philipp Schlicht. - 30 January: - cancelled
- 06 February: Ana Njegomir (Bonn) - Characterizations of some large cardinals

Itay Neeman introduced an approach to forcing with finite sequences of models of two types. We use this forcing to characterize some large cardinal properties through the validity of combinatorial principles in their forcing extensions. In particular, we will show, assuming that theta is regular and omega-inaccessible, that theta is inaccessible if and only if there are no weak Kurepa trees in the generic extensions. Also, we will show that assuming that kappa is inaccessible, we have that kappa is Mahlo if and only if there are no special aleph2-Aronszajn trees in the generic extensions.

This is joint work in progress with Philipp Lücke.