## 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 1.008, 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

**08 April**Peter Holy (University of Bristol)*L-like Models with Large Cardinals and a Quasi-Lower Bound on the Consistency Strength of PFA*

- A very appealing aspect of large cardinals in set theory is that they can be used to provide an almost linear ordering on set theoretical (or: mathematical) principles in terms of their consistency strength. While many results in this direction have been obtained by the "inner model programme" for set theoretic principles relating (in terms of their consistency strength) to "smaller" large cardinals, there are seemingly severe barriers to extend this to "large" large cardinals. I will present an alternative approach involving the "outer model programme" and "quasi-lower bounds": I will introduce those concepts and provide a "quasi-lower bound result" concerning the consistency strength of (a certain fragment of) a particular set theoretic principle, the proper forcing axiom (PFA). This is joint work with Sy Friedman.

**15 April**Asger Törnquist (University of Kopenhagen)*Statements that are equivalent to CH and their Sigma-1-2 counterparts.*

- There is a large number of "peculiar" statements that have been shown over time to be equivalent to the Continuum Hypothesis, CH. For instance, a well-known theorem of Sierpinski says that CH is equivalent to the statement that the plane can be covered by countably many graphs of functions (countably many of which are functions of x, and countably many of which are functions of y.) What happens if we consider the natural Sigma-1-2 analogues of these statements (in the sense of descriptive set theory)? It turns out that then these statements are, in a surprising number of cases, equivalent to that all reals are constructible. In this talk I will give many examples of this phenomenon, and attempt to provide an explanation of why this occurs. This is joint work with William Weiss.

**22 April**Heike Mildenberger (Universität Freiburg)*The filter dichotomy and the semifilter trichotomy*

- In the talk I will sketch a forcing construction that separates the filter dichotomy from the semifilter trichotomy. The preservation of given dense sets in the condensation order of block sequences is an important step of the proof. For the evaluation of the forcing we use the strong Ramsey-theoretic properties of stable ordered union ultrafilters.

**29 April**Martin Grohe (RWTH Aachen)*The Graph Isomorphism Problem*

- The question of whether there is a polynomial time algorithm deciding whether two graphs are isomorphic has been a one of the best known open problems in theoretical computer science for more than forty years. Indeed, the graph isomorphism problem is one of the very few natural problems in NP that is neither known to be in P nor known to be NP-complete. The question is still wide open, but a number of deep partial results giving polynomial time algorithms for specific classes of graphs are known. After an introductory survey on the graph isomorphism problem, in my talk I will discuss connections between a basic combinatorial isomorphism algorithm known as the Weisfeiler-Lehman algorithm, logical definability, and structural graph theory.

**06 May**Philipp Schlicht (Universität Bonn)*Generalized Choquet spaces and groups*

- We introduce an analogue to Polish spaces for uncountable regular cardinals kappa with kappa^< kappa = kappa via a variant of the Choquet game of length kappa. There is a surjectively universal such space, and any two such spaces of size > kappa with no points which are the intersection of fewer than kappa open sets are kappa-Borel isomorphic. We consider the special case of generalized kappa-valued ultrametric spaces with the property that the intersection of any decreasing sequence of balls is nonempty and construct a family of universal Urysohn spaces. We then prove that the logic action of Sym(kappa) is universal for kappa-Borel measurable actions of Sym(kappa) with respect to equivariant embeddings. This is joint work with Samuel Coskey.

**13 May**Peter Koepke (Universität Bonn)*Namba-like singularizations of successor cardinals*

- Bukowski-Namba forcing preserves aleph_1 and changes the cofinality of aleph_2 to omega. We lift this to cardinals kappa > aleph_1: Assuming a measurable cardinal lambda we construct models over which there is a further "Namba-like" forcing which preserves all cardinals <= kappa and changes the cofinality of kappa^+ to omega. Cofinalities different from omega can also be achieved by starting from measurable cardinals of sufficiently strong Mitchell order. Using core model theory one can show that the respective measurable cardinals are also necessary. This is joint work with Dominik Adolf (Münster).

*No talk on 20 May (holiday).*

**27 May***no talk*

**03 June**Katherine Thompson (Technische Universität Wien)*Oracle forcing in context*

**10 June**Merlin Carl (Universität Konstanz)*The distribution of ITRM-recognizable reals*

- Infinite Time Register Machines are a model of transfinite computation introduced in 2008 by P. Koepke and R. Miller. A real r is said to be ITRM-recognizable iff there is an oracle ITRM-program P such that P^x stops with output 1 iff x=r and otherwise with output 0. We will give a detailed picture how the ITRM-recognizable reals are distributed in Gödel's constructible universe L.

**17 June**Menachem Magidor (Einstein Institute of Mathematics, The Hebrew University of Jerusalem)*On compactness properties of successors of singular cardinals*

**24 June**Philipp Lücke (Universität Bonn)*Specializing Aronszajn trees and square sequences by forcing*

**01 July**Philipp Lücke (Universität Bonn)*Specializing Aronszajn trees and square sequences by forcing, continued*

**08 July**Mathias Beiglböck (Universität Bonn)*Ultrafilters in Additive Combinatorics*

- It is possible to extend an invariant mean on the integers to an invariant measure on the Stone-Cech compactification. We describe some applications which this fact has in additive number theory. We also compare the approach to alternative tools coming from ergodic theory.

**11 July**Thursday at 16.15 in room N0.008 (back building) in the Master seminar: Julian Schlöder (Universität Bonn)*Forcing axioms through iterations of minimal counterexamples*

**15 July**Robert Lubarsky (Florida Atlantic University)*Realizability Models Separating Various Fan Theorems*