# Oberseminar Mathematische Logik Wintersemester 2011

Dies ist die Homepage des Oberseminars "Mathematische Logik" im Wintersemester 2011.

### Zeit und Ort

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

• 26.10.2011
Adrian Mathias
Title: The use of ideas of Vopenka and Solovay to simplify definitions of sets of reals.
Abstract: Scott and Myhill in the 1960s developed the notion of ordinal definability, and defined the inner model HOD. Vopenka then showed that every real number is generic over HOD. Solovay's analysis of Lebesgue measurability led to the notion of an infinity-Borel set of reals. The work of the Californian Cabal on the axiom of determinacy has yielded a proof that if V=L(R) and AD holds, then every OD set of reals is infinity-Borel with code in HOD, In this talk, I shall present in some detail the above definitions and results, which play a pivotal role in Woodin's proof that AD yields inner models with many Woodin cardinals.

• 02.11.2011
Heike Mildenberger
Title: Specialising Aronszajn trees in a gentle manner
Abstract: We show that there are  proper forcings based upon countable trees of creatures that specialise a given Aronszajn tree and iterate these forcings with countable support. With a strong form of halving we ensure that the Ostaszewki club holds in the extension. This is joint work with Shelah.

• 9.11.2011
kein Vortrag

• 16.11.2011
Martin Ziegler
Titel: "Die Automorphismusgruppe des Urysohnraums"
Zusammenfassung: Wenn man den Normalteiler der beschränkten Automorphismen herausdividiert, ist die Automorphismengruppe einfach. Das ist eine gemeinsame Arbeit mit Katrin Tent.

• 23.11.2011
Mihai Prunescu
Title: $F_p$-affine recurrent double sequences oever $F_q$ are $p$-automatic
Abstract: A recurrent double sequence $a(m,n)$ is given by fixing two sequences $a(m, 0)$ and $a(0, n)$ as initial conditions and a rule of recurrence $a(m,n) = f(a(m, n-1), a(m-1, n-1), a(m-1, n))$ for $m, n \geq 1$. We show that if the sequences are $p$-automatic and have values in a finite field $F_q$ of characteristic $p$, and if the recurrence $f$ is $F_p$-affine, then the double sequence $a(m,n)$ is $p$-automatic. Consequently all such double sequences can be defined by matrix substitution. For the proof we show that the formal series $\sum a(m,n) X^m Y^n$ is algebraic over the field of rational functions $F_q(X, Y)$.

• 30.11.2011
Michael Pinsker
Title: Reducts of Ramsey structures: the canonical approach
Abstract: Consider the following problem: We are given a countably infinite relational base structure S; the goal is to determine its reducts, i.e., all relational structures T on the domain of S which have a first-order definition in S. When doing so, we identify reducts T, T' of S when they are first-order interdefinable.
This problem has been studied for numerous base structures S. For example, Cameron showed that the dense linear order has 5 reducts, Thomas proved that the random graph has 5 reducts as well, and Junker and Ziegler found that the dense linear order with a constant has 116 reducts.
With the original goal of refining Thomas' classification of the reducts of the random graph, Bodirsky and I developed a general method that turns out to be applicable to a considerable class of base structures S. For example, the method has since been used in order to find the reducts of the random partial order (number of reducts: 5 again!) and the random triangle-free graph with a constant (number of reducts: 13).
I will present our method and outline how it can be used in oder to reprove Thomas' theorem.

• 7.12.2011
Luca Motto Ros
The descriptive set-theoretical complexity of the embeddability relation
Abstract. We investigate the complexity of the embeddability relation on $$\mathrm{Mod}^\kappa_\varphi$$, the class of models of size $\kappa \geq \omega$ satisfying a given $\mathcal{L}_{\kappa^+ \kappa}$-sentence $$\varphi$$, in terms of some reducibility notions which are very popular in descriptive set-theory. In particular, we will show that if either $$\kappa = \omega$$ or $$\kappa$$ is weakly compact, then the embeddability relation is \emph{(invariantly) universal}, i.e.\ for every analytic quasi-order $$R$$ on $$\pre{\kappa}{2}$$ there is an $$\L_{\kappa^+ \kappa}$$-sentence $$\varphi$$ such that $$R$$ is Borel bi-reducible with the relation of embeddability on $$\mathrm{Mod}^\kappa_\varphi$$.

• 14.12.2011
Viktor Selivanov
Title: A Gandy Theorem for Abstract Structures and Applications to First-Order Definability
Abstract: We establish a Gandy theorem for a class of abstract structures and deduce some corollaries, in particular the maximal definability result for arithmetical structures in the class. We also show that the arithmetical structures under consideration are biinterpretable (without parameters) with the standard model of arithmetic. As an example we show that for any k3 a predicate on the quotient structure of the h-quasiorder of finite k-labeled forests is definable iff it is arithmetical and invariant under automorphisms. This structure is closely related to the extension of the Wadge reducibility to k-partitions. There are also other natural examples of such structures. This is a joint work with Oleg Kudinov.

• 21.12.2011
Hans Adler
Titel: Theorien ohne die Baumeigenschaft 2. Art
Zusammenfassung: Theorien ohne die Baumeigenschaft 2. Art verallgemeinern sowohl einfache Theorien als auch Theorien ohne die Unabhängigkeitseigenschaft. Einige Ergebnisse der letzten Zeit, insbesondere Kim's Lemma für Theorien ohne die Baumeigenschaft 2. Art, legen es nahe, dass man auch in diesem Kontext noch Stabilitätstheorie treiben kann.

• 11.01.2012
Riccardo Camerlo
Title: The density point property
Abstract: Some results are presented about spaces that satisfy (or do not satisfy) the density point property. In particular, a thorough investigation is performed on the Cantor space.

• 18.01.2012
Martin Ziegler
Titel: "Generische Automorphismen von abzählbaren Strukturen"
Zusammenfassung: Wir geben einen einfachen Beweis für das Kechris-Rosendahl-Kriterium für die Existenz von generischen Automorphismen abzählbarer Strukturen.

• 25.01.2012
Juan-Diego Caycedo
Title: Tame expansions of the real and complex fields
Abstract: The complex field and the real ordered field are prototypical examples of well-behaved structures from the model-theoretic viewpoint, although in rather different ways. Indeed, in both cases there is a certain geometric understanding of definable sets. Natural questions are what expansions of these structures introducing new definable sets admit similar results and where is there a dividing line between "tame" and "wild" expansions.
In the case of the real field, a rich theory has been developed for o-minimal expansions. Also, a meaningful division is obtained when tame expansions are defined to be those which do not define the set of integer numbers.
For expansions of the complex field, it is less clear what the notion of tame expansions should be. In this context, I will discuss some examples of structures related to complex exponentiation and some relations with the real case.

• 01.02.2012
kein Vortrag

• 8.2.2012
Heike Mildenberger
Determiniertheit und Forcing
Mit einer Definition der Forcingrelation mittels Gewinnstrategien in einem unendlichen Zweipersonenspiel beweisen wir den Satz von Donald Martin, dass Delta_1^1-Spiele im Raum $\lambda^\omega$ determiniert sind.

• 15.02.2012
Luis Miguel Villegas Silva
Title: The two-cardinal transfer problem: the singular case
Abstract: Let κ , λ be inﬁnite cardinals, λ be singular. Let L be a ﬁrst order countable language with at least one symbol of unary predicate U. Let A be an L-structure of type (κ ++ , κ ), that is, |A| = κ ++ and |U A | = κ . We shall prove that there exists an L-structure B of type (λ ++ , λ ) elementarily equivalent to A.
This problems was solved by Jensen in the late 70’s but unfortunately the manuscript of the proof is lost. Jensen asked me to provide a complete proof of this case of the two-cardinal transfer problem.
© May 2008 H.M., last update on January 26, 2012