## Oberseminar mathematische Logik

### Organizers

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

### Time and location

Monday 16.30-18.00 in room 0.011, 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.

### Contents

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

### Plan

**14 April**Peter Koepke (Universität Bonn)*Felix Hausdorff and the Foundations of Mathematics*

- Exactly 100 years ago, after a long phase of foundational uncertainty Felix Hausdorff's "Grundzüge der Mengenlehre" (Foundations of Set Theory) established set theory as a comprehensive field of mathematics. Hausdorff advocated set theory as a universal foundation of mathematics. He followed David Hilbert's axiomatic method and formalism. Hausdorff's position corresponds closely to the anti-metaphysical stance in his philosophical book "Das Chaos in Kosmischer Auslese" (Chaos in Cosmic Selection). A formalist like Hausdorff selects consistent axiom systems from the chaos of mathematical possibilities, guided by various criteria. Not least by intellectual and aesthetics considerations.

**21 April***no talk*

**05 May**Daniel Kuehlwein

*Machine Learning for Automated Reasoning*

- I will give a summary of my PhD research on applying machine learning to improve automated reasoning systems. The focus will be (mainly) on the premise selection problem: Given a set of premises (e.g. library of axioms, definitions and already proved theorems) and a new conjecture, predict which premises are useful to prove the conjecture. The results of this research have been integrated in the ITP Isabelle. On average, 70% of the problems can now be solved fully automatic.

**12 May***no talk*

**26 May**Vladimir Kanovei (Moscow)

*On the automorphisms in the Gitik-Koepke construction*

- It is known that the assumption that GCHfirst fails at aleph0 implies in ZFC the existence of inner models with large cardinals. Gitik and Koepke demonstrated that this is not so without the axiom of choice. Namely there is a cardinal-preserving symmetric-generic extension of L, in which GCH holds at every cardinal aleph n but there is a surjection from the power set of aleph omega onto any previously chosen cardinal in L, as large as one wants, and the axiom of choice by necessity fails. In other words, in such an extension GCH holds in the proper sense for all cardinals aleph n but fails at aleph omega in Hartogs' sense. The goal of this talk is to analyse the system of automorphisms involved in the Gitik-Koepke construction.

**30 May**Friday, 16.30-18.00, room 1.007 Vassilis Gregoriades (Darmstadt)*A recursive theoretic view to the decomposability conjecture*

- The decomposability conjecture states that every function from an analytic space to a separable metric space, for which the preimage of a Σ^0_{m+1} set is a Σ^0_{n+1} set, where m=1,2,...n, is decomposable into countably many Σ^0_{n-m+1}-measurable functions on Π^0_n domains. The aim of this talk is to present some recent results about this problem in zero-dimensional spaces. This is a joint work of Kihara and the speaker. The proofs make use of results from recursion theory and effective descriptive set theory, including a lemma by Kihara on canceling out Turing jumps and Louveau separation. We will first review the necessary material and then we will proceed to the proof of the new results. Moreover we will explain how these results can be extended from the context of zero-dimensional spaces to spaces of small inductive dimension.

**02 June***no talk*

**09 June***no talk*

**16 June**Mirna Dzamonja (Norwich)*Embeddings of graphs with no large cliques*

- We shall discuss embeddings between graphs omitting large cliques and in particular we shall prove that for kappa singular of cofinality kappa there is no universal graph of size kappa omitting cliques of size kappa, just in ZFC.

**23 June**Otmar Spinas (Kiel) "Das Problem mit Silver Amoeba"

*7 July no talk*

**14 July**Giorgio Laguzzi (Hamburg)

*Roslanowski and Spinas dichotomies*

- We investigate two tree forcings for adding infinitely often equal reals: the full splitting Miller forcing FM, introduced by Roslanowski, and the infinitely often equal trees forcing IE, implicitly introduced by Spinas. We prove results about Marczewski-type regularity properties associated with these forcings as well as dichotomy properties for projective sets, with a particular emphasis on a parallel with the Baire property. This is joint work with Yurii Khomskii.