## Oberseminar mathematische Logik

### Organizers

- Prof. Dr. Stefan Geschke
- Prof. Dr. Peter Koepke
- Dr. Philipp Schlicht

### Time and location

Monday 16.30-18.00 in room 1.007, Endenicher Allee 60.

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

### Contents

Advanced talks on mathematical logic by guests and members of the logic group.

### Plan

**15 October**Philipp Lücke*Free groups and automorphism groups of infinite fields*

**22 October**Philipp Schlicht*Perfect subsets of generalized Baire spaces*

**29 October**Stefan Geschke*An induced Ramsey theorem for clopen graphs*

- We use forcing to prove Ramsey theorems for colorings of closed subsets of large clopen graphs.

**5 November**David Schrittesser (Wien)*Separating ideals and Jensen Coding with Easton Support*

**3 December**Dominik Adolf (Universität Münster)*The strength of PFA(aleph_2) plus a precipitous Ideal on omega_1*

- We will present an argument for extracting strength from the above mentioned hypothesis building on a work of Claverie and Schindler. We will give a short overview of the inner model theory used with a focus on showcasing the core ideas underpinning our methods. We will discuss limitations of the argument and ideas on how to overcome them.

**5 December**(Wednesday) Zum Dies Academicus trägt Peter Koepke um 15.15 Uhr im Hörsaal IV im Hauptgebäude vor:*Berechenbarkeit und ihre Grenzen - zum 100. Geburtstag von Alan Turing*

**17 December**Luca Motto Ros (Universität Freiburg)*The embeddability relation on models of size kappa is (strongly) invariantly universal when kappa^{<kappa}=kappa*

**07 January**Marcin Sabok (Polish Academy of Sciences, Warsaw)*A dichotomy in canonization of analytic equivalence relations*

- I will show and discuss a dichotomy that occurs in attempts of canonization of analytic equivalence relations. The dichotomy says that under suitable minimality condition for P_I, if E is an analytic equivalence relation on a Polish space, then - either there is a Borel I-positive set which consists of E-independent elements (i.e. E canonizes to the identity) - or there is a Borel I-positive set such that E|B is I-ergodic (this means the saturations of every two Borel I-positive subsets of B have nonempty intersection ) I will present some typical examples for both types of behavior and sketch a proof of the dichotomy. The proof will use suitably chosen generic ultrapowers. This is joint work with Vladimir Kanovei and Jindra Zapletal.

- We show that in Zermelo-Fraenkel set theory without the Axiom of Choice, a surjectively modified continuum function can take almost arbitrary values.

**21 January**Mona Rahn will give a talk in the Master student's seminar (16.30 in room 1.007):*Hat games without the axiom of choice*

- Consider the following game: omega many people named 0, 1, 2, . . . sit in a row, where person n sees every person with a greater number, namely n+1, n+2, etc. Each person wears a hat that has a color taken from a fixed set of colors. Starting with the person who sees all the others, the people guess the color of their hats, one after the other. The people are allowed to agree on a strategy beforehand but are not allowed to communicate once they wear their hats. However, they do hear the previous guesses. Surprisingly, assuming the Axiom of Choice, there is always a strategy that guarantees that at most one person guesses his hat wrong. We investigate what happens when we let go of the Axiom of Choice. We present a model of ZF showing that AC is necessary in order to find the aforementioned strategies for at most countable sets of colors. This is due to the fact that the existence of those strategies implies, among other things, the existence of a set of reals without the Baire property and a subset of [omega]^omega that is not Ramsey.

**28 January**Lutz Strüngmann (Hochschule Mannheim)*Endomorphisms of aleph_n-free modules over the p-adic intergers*