# Oberseminar Mathematische Logik Wintersemester 2014/15

 Mathematisches Institut Abteilung für Math. Logik

Dies ist die Homepage des Oberseminars "Mathematische Logik" im Wintersemester 2014/15.

Prof. Ziegler hat ein Forschungssemester.

### Zeit und Ort

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

### Vorträge

• October, 22
Giorgio Laguzzi
Between Sacks and random for inaccessible kappa
Abstract: A non-trivial issue concerning tree-like forcings in the generalized framework is to introduce a random-like forcing, where random-like means to be κκ-bounding, < κ-closed and κ+-cc simultaneously. Shelah managed to do that for κ weakly compact. In this talk we aim at introducing a forcing satisfying these three properties for κ inaccessible, and not necessarily weakly compact. This is joint work with Sy Friedman.
• October, 29
Heike Mildenberger
Reflection principles
This is an introduction to reflection principles, on how to force them and on their consistency strength. This is based on work by Justin Moore and Stevo Todorcevic.
• November, 5
Christoph Bier
Basic Notions: Forcing
• November, 12
David Chodounsky
Y-c.c. and Y-proper forcings
I will introduce a new type of properties of c.c.c. and proper forcing notions, of which the Y-c.c. and Y-proper are the two most prominent examples. The Y-c.c. is an intermediate property between σ-centered and c.c.c., and Y-proper is intermediate between strongly proper and proper. These properties have interesting consequences and behave nicely with respect forcing iterations.
• November, 19
Philipp Lücke
Representing sets of cofinal branches as continuous images
Let $\kappa$ be an infinite cardinal and $T$ be a tree of height $\kappa$. We equip the set $[T]$ of all branches of length $\kappa$ through $T$ with the topology whose basic open subsets are sets of all branches containing a given node in $T$. Given a cardinal $\nu$, we consider the question whether $[T]$ is equal to a continuous image of the tree of all functions $s:\alpha\longrightarrow\nu$ with $\alpha<\kappa$. This is joint work with Philipp Schlicht.
• November, 26
Ilya Sharankou
Overview of the generalized combinatorial cardinal characteristics
I will talk about how the combinatorial cardinal characteristics reviewed in [Blass, Combinatorial Cardinal Characteristics of the Continuum] can be generalized to uncountable cardinals kappa and what is known about consistency results for them.
• December, 3
Otmar Spinas
Silver trees and Cohen reals
I will sketch the main ideas of my recent result that the meager ideal is Tukey reducible to the Mycielski ideal. The latter one is the ideal associated with Silver forcing. This implies that every reasonable amoeba forcing for Silver adds a Cohen real. This has been open for some years.
• December, 10
Lothar Krapp
Schanuel's Conjecture and Exponential Fields
Schanuel's Conjecture states that for a collection of n complex numbers z_1, ..., z_n, linearly independent over the field of rational numbers, the transcendence degree of z_1, ..., z_n, exp(z_1), ..., exp(z_n) is at least n. Zilber constructs in [Zilber, Pseudo-exponentiation on algebraically closed fields of characteristic zero] a sentence whose models are structures called strongly exponentially-algebraically closed fields with pseudo-exponentiation, which are unique in every uncountable cardinality. One of their main properties is that Schanuel's Conjecture holds in those fields. Firstly, I will outline the properties of Zilber's fields. Secondly, I will sketch the proof given in [Marker, A Remark on Zilber's Pseudoexponentiation] showing that, if one assumes Schanuel's Conjecture, the simplest case of one of the axioms of Zilber's fields holds in the complex exponential field.
• December, 17
Giorgio Laguzzi
Amoeba and tree ideals
I will talk about what I asked to Spinas in the end of his talk, i.e., whether an amoeba for Silver might add Cohen reals. Two weeks ago he proved that add(J(Silver)) is at most add(M). However this is not strictly sufficient to infer that any proper amoeba for Silver does add Cohen reals, but only that it does not have the Laver property. I will clarify this issue. If there will be any time left I will also present some results about other tree ideals, which are part of a joint work, still in preparation, with Yurii Khomskii and Wolfgang Wohofsky.
• January, 7, 2015
Kein Oberseminar. Hausbegehung und Sitzungen im Rahmen des Akkreditierungsverfahrens.
• January, 14, 2015
Andrew Brooke-Taylor
Cardinal characteristics at supercompact kappa in the small u(kappa), large 2^kappa model
When generalising arguments about cardinal characteristics of the continuum to cardinals kappa greater than omega, one frequently comes up against the problem of how to ensure that a filter built up through an iterated forcing remains kappa complete at limit stages of small cofinality.  A technique of Dzamonja and Shelah is useful for overcoming this problem; in particular, there is a natural application of this technique to obtain a model in which 2^kappa is large but the ultrafilter number u(kappa) is kappa^+.  After introducing this model, I will talk about joint work with Vera Fischer (Technical University of Vienna) and Diana Montoya (University of Vienna) calculating many other cardinal characteristics at kappa in the model and its variants.
• January, 21, 2015
Juan-Diego Caycedo
On the theory of universal specializations of Zariski structures
I will report on ongoing joint work with Alex Berenstein and Alf Onshuus on proving that all universal specializations of models of the theory of a fixed Zariski structure share the same complete theory and finding axioms for this theory. I will show this in the most fundamental particular cases (sets, vector spaces, algebraically closed fields) and discuss our progress in the general case. I spoke about this topic once before in the seminar and the current talk is a continuation of the previous one, but no knowledge of the content of that talk will be assumed.
• January, 28, 2015
kein Oberseminar
• February, 4, 2015
Moritz Müller
Partially definable forcing and weak arithmetics
Given a nonstandard model M of arithmetic we want to expand it by interpreting a binary relation symbol R such that R^M does something prohibitive, e.g. violates the pigeonhole principle in the sense that R^M is a bijection from n+1 onto n for some (nonstandard) n in M. The goal is to do so saving as much as possible from ordinary arithmetic. More precisely, we want the expansion to satisfy the least number principle for a class of formulas as large as possible. We describe a forcing method to produce such expansions and revisit the most important results in the area.
• February, 11, 2015
Heike Mildenberger
A new forcing order
We will see a new type of expanded Sacks forcing with many combinatorial properties.
Last update on January 29, 2015, H.M.