# Oberseminar Mathematische Logik Sommersemester 2013

 Mathematisches Institut Abteilung für Math. Logik

Dies ist die Homepage des Oberseminars "Mathematische Logik" im Sommersemester 2013.

### Zeit und Ort

Mi 16:30-18:00, SR 404 in der Eckerstr. 1, vorher ab kurz nach 4 Tee in Zimmer 310

### Vorträge

• 17.4.2013
Luis Miguel Villegas Silva
The principle Diamond Star
Abstract: In this talk we shall present the combinatorial principle diamond(k) star. It is
stronger than usual diamond(k). It is known that
in L, diamond(k) star holds iff k is not ineffable. We will sketch the proof that diamond(k) star
holds when k is a limit cardinal not ineffable, under GCH.
• 24.4.2013
Martin Ziegler
Pseudoräume
• 1.5.2013
holiday
http://de.wikipedia.org/wiki/Erster_Mai
• 8.5.2013
Regula Krapf
The pseudointersection number and the tower number
• 15.5.2013
Adrian Mathias
Unsound Ordinals
An ordinal zeta is *unsound* if there are subsets A_n (n in omega) of it such that as b ranges through the subsets of omega, uncountably many ordertypes are realised by the sets $\bigcup_{n \in b} A_n$.

Woodin in 1982 raised the question whether unsound ordinals ordinals exist; the answer I found then (to be found in a paper published in the Mathematical Proceedings of the Cambridge Philosophical Society volume 96 (1984) pages 391--411) is this:

Assume DC. Then the following are equivalent:

i) the ordinal $\omega_1^{\omega + 2}$ (ordinal exponentiation) is unsound

ii) there is an uncountable well-ordered set of reals

That implies that if omega_1 is regular and the ordinal mentioned in i) is sound, then omega_1 is strongly inaccessible in the constructible universe. Under DC, every ordinal strictly less than the ordinal mentioned in i) is sound.

There are many open questions in this area: in particular, in Solovay's famous model where all sets of reals are Lebesgue measurable, is every ordinal sound ? The question may be delicate, as Kechris and Woodin have shown that if the Axiom of Determinacy is true then there is an unsound ordinal less than omega_2.

• 22.5.2013
Pfingstpause
• 29.5.2013
Heike Mildenberger
Resurrecting Ramsey ultrafilters
Suppose we have a Ramsey ultrafilter $V$ in the ground model and another, non-nearly coherent $P$-point $E$. Is there a forcing that destroys $V$, preserves $E$ (and hence $\aleph_1$), such that in the generic extension $V$ can be complemented to a Ramsey ultrafilter again? Using work by Blass (1987) and Eisworth (2002) and of my own we show that there is a forcing that destroys $V$, keeps $E$, and forces that in the extension $V^+$, the set of $V$-positive sets, is a selective coideal. Then we can cite a proposition of Adrian Mathias (Happy Families, Annals of Math. Log (1977)) and get in the extension a Ramsey ultrafilter $V_1 \subseteq V^+$. I will also explain the purpose of such technical work.
• 5.6.2013
Giorgio Laguzzi
• Generalized random forcing for weakly compact
When dealing with the generalized Cantor space 2^\kappa, for \kappa regular uncountable, the notion of Cohen forcing and Baire property trivially generalizes. What is more complicated to do is to find a right generalization of random forcing, since the usual notion of Lebesgue measure does not generalizes straightforwardly. During the talk, we present Shelah's recent result shedding light on this problem, by introducing a forcing which is simultaneously \kappa^+-cc and \kappa^\kappa-bounding, for \kappa weakly compact.

• 12.6.2013
Jörg Flum
On Limitations of the Ehrenfeucht-Fraïssé method
Ehrenfeucht-Fraïssé games and their generalizations have been quite successful in finite model theory and yield various inexpressibility results. However, for key problems such as the P- NP- problem no progress has been achieved using the games. We show that for these problems it is already hard to get the board for the corresponding Ehrenfeucht-Fraïssé game. We obtain similar results for the so-called Ajtai-Fagin games and for a variant where the structures are obtained randomly.
• 19.6.2013
Markus Junker
Heyting algebras
• 26.6.2013
Menachem Magidor
Inner Models for Set Theory Defined by Generalized Logics
pdf file of the abstract
• 3.7.2013
Misha Gavrilovich
A category-theoretic viewpoint on first definitions in general topology
We observe that several definitions in a first course on general topology, such as Hausdorff, dense, T_0, T_1, admit an easy reformulation as computations with partial preorders of category-theoretic nature. Namely, these computations correspond to rules for manipulating commutative diagrams involving only finite topological spaces as constants (and variables). We suggest a calculus based on these rules.
• 10.7.2013
Juan Diego Caycedo
The real field with dense subgroups of the torus
• 17.7.2013
Fourth European Set Theory Conference, Barcelona
Last update on May 8, 2013, H.M.