## Oberseminar Mengenlehre - Graduate seminar on set theory

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

*Master module S4A3, 6 credit points, workload 180h*

## Organized by

## Prof. Dr. Peter Koepke

### Time and location

Lectures: Tuesdays 16:30-18:00 (SR B)

The members of the seminar are welcome for coffee and tea at
Peter Koepke's office, Room 44, Beringstrasse 4
from 16:00-16:30 before the talks.

### Program

23 October,

Lutz Strüngmann(Essen-Duisburg):"Die Struktur von Ext(G,H)"In 1977 Saharon Shelah solved the well-known Whitehead problem by showing that it is undecidable in ordinary set-theory ZFC wether or not every abelian group G satisfying Ext

_{Z}(G,Z)={0} has to be free. However, this did not clarify the structure of Ext_{Z}(G,Z) for torsion-free abelian groups - a problem which has received much attention since then. Easy arguments show that Ext_{Z}(G,Z) is always a divisible group for every torsion-free group G. Hence it is of the formExt

_{Z}(G,Z)= ⊕_{p ∈ Π}Z(p^{∞})^{(νp)}⊕Q^{(ν0)}for some cardinals ν

_{p}, ν_{0}(p ∈ Π) which are uniquely determined. The obvious question that arises is which sequences (ν_{0}, ν_{p}: p ∈ Π) of cardinals can appear as the cardinal invariants of Ext_{Z}(G,Z) for some (which) torsion-free abelian group? We will give a complete characterization assuming Goedel's constructible universe L plus there is no weakly compact cardinal. Moreover, we shall consider different models of ZFC in which we can reach the borderline. For instance, we use the existence of a super compact cardinal to show that there is a model of set theory in which the cardinal invariants of any torsion-free group are as amximal as possible. Other models arecloseto ZFC but still we have a strange behaviour of those invariants. All of this is joint work with Saharon Shelah. Last but not least we will consider related structure problems for the functor Bext which plays an important role in the theory of infinite rank Butler groups. This is work jointly done with Rüdiger Göobel and Nicole Hülsmann.29 October, No Oberseminar this week; instead we attend the

Festkolloquium zum 65. Geburtstag von Stuhlmann-LaeiszHauptgebäude Festsaal der Universität Bonn.

Programm:

16.15 Uhr -- 17.15 Uhr

Prof. Dr. Ulrich Nortmann (Saarbrücken):

Kunst, Erkenntnis und Erkenntnis durch Kunst

17.15 Uhr -- 17.30 Uhr

Festansprache und musikalische Darbietung

17.30 Uhr -- 18.30 Uhr

Prof. Dr. Andreas Bartels (Bonn):

Ein wissenschaftstheoretischer Blick auf die

Kosmologie

6 November,Katie Thompson(Vienna):"How to achieve Global Domination (in an inner model)"The dominating number for κ, denoted d(κ), is the least size of a family of functions from κ to κ which eventually dominates the rest. Global Domination is the statement d(κ) < 2

^{κ}for all regular κ. I will talk about how to build a class forcing which achieves Global Domination in such a way that a class generic may be built assuming O^{#}exists. This is done in the context of the inner model program started by Sy Friedman, one of the aims of which is to discover what may be true in an inner model with minimal large cardinal assumptions. The results in this talk are joint work with Sy Friedman.

13 November,Brian Semmes(Amsterdam):"Games, trees, and Borel functions"In this talk, I will give an overview of my thesis work: a game theoretic analysis of various function classes on the Baire space.

20 November,

Vladimir Kanovei(Moscow):"Lebesgue measure and the coin-tossing game"Given a set A of infinite dyadic sequences, we consider a game between G, the gambler, and C, the casino. C successively plays bits b

_{0},b_{1},b_{2},... , and C definitely loses if the infinite sequence b=<b_{0},b_{1},b_{2},...> does NOT belong to A. And G bets on every next move of C. Beginning with the initial balance say 1, G can bet any amount less than the current balance on one of two possible moves of C (0 or 1), and if C makes that move then the balance accordingly increases by the amount of bet. Otherwise the balance decreases. The final outcome of the game can be defined in terms of the limit of the supremum of the balance values. And it turns out that the existence of certain strategies for G and C characterizes the Lebesgue measure characteristics of the set A. In brief, the smaller A is the bigger gains Casino can guarantee.27 November,

Andy Lewis(Leeds):"On the degree spectrum of a Π^{0}_{1}class"Peter asked me to give a talk that would be of interest to those of you who have been doing some work in ordinal computability. Not working in this area, I will not discuss the subject directly, but will talk instead about some areas in classical computability for which it may be interesting to ask what changes when the classical Turing model is substituted, for example, for that of infinite time Turing machines. I shall begin by giving an introduction to the study of Π

^{0}_{1}classes from a degree theoretic point of view. This is an area which is very basic to classical computability, and very little previous understanding of computability theory will be required. After this introduction I will then go on to discuss some questions in this area with which I have been concerned over the last couple of months, and which have brought to light some interesting phenomena relating to the degree spectra of Π^{0}_{1}classes. In this second part of the talk, then, we shall mainly be concerned with the structure which is defined by ordering the degree spectra of Π^{0}_{1}classes by inclusion, but we will also be distracted by some issues of independent interest that arise along the way.Monday 10 December,

Russel Miller(New York):"Computable Structures and Computable Categoricity"Computable model theory asks to what extent constructions in model theory can be performed effectively. When considering categoricity, therefore, we wish to know to what extent computable isomorphisms exist between various structures: a computable structure

Sis said to becomputably categoricalif every computable structure which is classically isomorphic toSis in fact computably isomorphic toS. (Notice that this is a property of structures, not of theories.) We will begin with simple examples and continue to computable trees and fields, asking which ones are computably categorical and examining two different approaches to the problem.

Some of the work on trees is joint with Lempp, McCoy, and Solomon, and some of the work on fields is joint with Schoutens.

29 January,

Bernhard Irrgang(Bonn):"Remarks on a negative partition relation"