# Oberseminar Mathematische Logik Sommersemester 2012

Dies ist die Homepage des Oberseminars "Mathematische Logik" im Sommersemester 2012.

### Zeit und Ort

Mi 16:45-18:00, SR 404 in der Eckerstr. 1, vorher Tee in Zimmer 310

• 24.04.2012
Title: Morleys Kategorizitätssatz und die Klassifikation streng-minimaler Mengen

• 09.05.2012
Marcin Sabok
Title: Canonization of hypersmooth equivalence relations
Abstract: I will discuss some recent developments in canonical Ramsey
theory in the context of descriptive set theory. Given a class
$\mathbf{E}$ of Borel equivalence relations and a sigma-ideal $I$ on a
Polish space $X$, we say that the class_canonizes_  to a finite set
$\{F_1,\ldots,F_n\}$ of equivalence relations if every Borel
equivalence relation in the class $\mathbf{E}$ is equal to one of
$F_1,\ldots,F_n$ after a restriction to a Borel I-positive set. I will
discuss a canonization result for the class of hypersmooth equivalence
relation on the space $\mathbb{R}^\omega$. Let $I$ be the family of
Borel subsets of $\mathbb{R}^\omega$ on which $E_1$ is hyperfinite.
Then every hypersmooth equivalence relation is equal to the three
equivalence relations: identity, everything or $E_1$ after a
restriction to an $I$-positive Borel set. If time permits, I will
sketch a proof, which uses techniques coming from proper forcing and
the reverse iteration of Sacks forcing. This is joint work with

• 23.05.2012
Luca Motto Ros
Title: On an old question of Lusin concerning countably continuous Borel functions
Abstract: A famous question of Lusin asked whether every Borel function is countably continuous, i.e. can be written as a countable union of partial continuous functions (with arbitrary domains). This question can be straightforwardly generalized by replacing continuous functions with functions of a fixed Baire class. Both questions where answered negatively already in the Thirties, but (except for the basic case of countably continuous functions, where the Pawlikowski function provides a somewhat canonical counterexample) the desired functions are usually obtained indirectly using nontrivial Baire category argument, universal functions and diagonalization. The aim of this talk is to present new extremely simple counterexamples, which are in a sense canonical and can be seen as generalizations of the Pawlikowski function. Such examples allow also to fully describe the structure under inclusion of finite level Borel classes of functions. Finally, if time permits we will also present some (partial) positive results showing that functions appearing in certain finite level Borel classes turn out to be always countably continuous in a definable way.

• 13.06.2012
Matteo Viale
Title: Forcing and absoluteness as means to prove theorems
Abstract: The forcing method has been introduced by Cohen in the early sixties to
prove the independence of the continuum problem. Forcing can be presented
as an “algorithm” which takes as inputs a model M of ZFC and a boolean
algebra B in M and produces a boolean valued model M^B of ZFC. The
first order theory of M^B depends on the first order theory of M and on
the combinatorial properties of B. Since its introduction forcing has been
the most powerful tool to prove independence results in set theory. In
this talk we shall take a dIfferent attitude and show that forcing is a
powerful tool to prove theorems in ZFC.

Those interested in the argument of this seminar can consult the preprint
Martin’s maximum revisited:
http://www2.dm.unito.it/paginepersonali/viale/Martinmaximumrevisited.pdf

• 04.07.2012
Yijia Chen
Title: Levin`s optimal inverters

• 11.07.2012
Tagung Trends in Set Theory in Warschau

• 18.07.2012
Jörg Flum
Title: On the ordered conjecture
Abstract: In a paper of 1978 Podewski and Ziegler showed that graphs satisfying a certain combinatorial property are stable. In the talk we present some generalizations of a corollary of this.

© May 2008 H.M., last update on July 13, 2012