## Oberseminar mathematische Logik

### Organizers

- Dr. Peter Holy
- 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

- 23 October: Peter Holy (Bonn) The exact strength of the class forcing theorem

We consider second order set theories, that have as objects both sets and classes, and the role of the class forcing theorem, that is the forcing theorem for all notions of class forcing, within this range of theories. While Kelley-Morse class theory (KM) proves the class forcing theorem, its failure is consistent with the axioms of Gödel-Bernays set theory (GBC). We show that the class forcing theorem is equivalent, over GBC, to the principle of elementary transfinite (class) recursions of length Ord, and to the existence of various kinds of truth predicates. This is joint work with Victoria Gitman, Joel Hamkins, Philipp Schlicht and Kameryn Williams.

- 06 November: Alexey Ostrowski (St. Petersburg) Around the problem of Lusin

We consider problems related to preservation of classes of subsets of separable metric spaces under continuous images and partitions of functions into countably many continuous functions. This continues work of Hausdorff, Lusin, Novikov and the author.

- 13 November: Philipp Lücke (Bonn) Squares, chain conditions, and products

With the help of square principles, we obtain results concerning the consistency strength of several statements about strong chain conditions and their productivity. In particular, we show that if the κ-Knaster property is countably productive for some uncountable regular cardinal κ, then κ is weakly compact in L. The proof of this result relies on a new construction that shows that Todorcevic’s principle □(κ) implies an indexed version of the principle □(κ,λ). This is joint work with Chris Lambie-Hanson (Bar-Ilan).

- 20 November: Philipp Schlicht (Bonn) The Hurewicz dichotomy for definable subsets of generalized Baire spaces

By classical results of Hurewicz and Saint-Raymond, an analytic subset of a Polish space X is covered by a Ksigma subset of X if and only if it does not contain a closed subset of X that is homeomorphic to the Baire space. Moreover, Kechris proved that this result generalizes to the projective sets if projective determinacy is assumed. We consider the analogous statement - the Hurewicz dichotomy - for subsets of the generalized Baire space kappa^kappa for a given uncountable cardinal kappa with kappa^(<\kappa)=kappa. We sketch a proof of the consistency of the Hurewicz dichotomy for all subsets of the generalized Baire space kappa^kappa that are definable from parameters in kappa^kappa. This is joint work with Philipp L\"ucke and Luca Motto Ros.

- 27 November: Merlin Carl (Konstanz) Complexity theory for ordinal Turing machines

Ordinal Turing Machines (OTMs) generalize Turing machines to transfinite working time and space. We consider analogues of theorems from complexity theory for OTMs, among them the Cook-Levin theorem, the P vs. NP problem and Ladner's theorem. This is joint work with Benedikt Löwe and Benjamin Rin.

- 04 December: Daniel Soukup (Vienna) Strongly surjective linear orders

A linear order L is strongly surjective if L can be mapped onto any of its suborders in an order preserving way. Our goal will be to review various recent results on the existence and non-existence of uncountable strongly surjective linear orders answering questions of Camerlo, Carroy and Marcone. In particular, I would like to sketch the proof that in J. Moore's model of CH + axiom A, any strongly surjective linear order is countable.

- 11 December: Aleksandra Kwiatkowska (Münster) Universal minimal flows of Ważewski dendrites

We study universal minimal flows of the homeomorphism groups of Ważewski dendrites W_P, where P\subset {3,4,...,\omega}. If P is finite, we prove that the universal minimal flow of Homeo(W_P) is metrizable and we compute it explicitly. This answers a question of B. Duchesne. If P is infinite, we show that the universal minimal flow of Homeo(W_P) is not metrizable. This provides examples of topological groups which are Roelcke precompact and have a non-metrizable universal minimal flow.

- 18 December: Dan Nielsen (Bristol) Mapping the Ramsey-like cardinals

Ramsey-like cardinals were introduced in Gitman (2011) and Gitman & Welch (2011), broadly speaking being cardinals k that are critical points of elementary embeddings from a size k ZFC^- model. Recently, Holy & Schlicht (2017) have introduced a new large cardinal into the Ramsey-like family, called (strategic) alpha-Ramsey cardinals, whose distinctive feature is that they admit a game-theoretic characterisation. I will present some new results concerning how these Ramsey-like cardinals fit into the large cardinal hierarchy and how they interact with the core model K. This is joint work with Philip Welch.

- Gitman, Victoria: "Ramsey-like cardinals", JSL 76 (2011), no. 2, 519-540

- Gitman, Victoria and Welch, Philip D.: "Ramsey-like cardinals II", JSL 76 (2011), no. 2, 541-560

- Holy, Peter and Schlicht, Philipp: "A hierarchy of Ramsey-like cardinals", preprint (2017), arXiv:1710.10043.

- 08 January: no talk because of the Toeplitz Kolloquium

- 29 January: Farmer Schlutzenberg (Münster) Semiscales constructed directly from mice