## 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 Todorcevicâ€™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 (Constance) Complexity theory for ordinal Turing machines

- 04 December: Daniel Soukup (Vienna) tba

- 11 December: Aleksandra Kwiatkowska (Münster) tba

- 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.

- 09 January - Tuesday 16.30-18.00 in room N0.008: Yuri Khomskii (Hamburg) tba

- 15 January: tba

- 22 January: tba

- 29 January: tba