## Oberseminar Mathematische Logik

### Time and Place

Tuesdays 16:30-18:30 Beringstraße 4 SR A

### Program

1. #### Gunter Fuchs (Münster): Maximalitätsprinzipien für abgeschlossene Forcings

April 10, 2007
Zusammenfassung: Ich untersuche Varianten der Forcing-Maximalitätsprinzipien, die Joel Hamkins eingeführt hat, für Klassen von kappa-abgeschlossenen forcings, für verschiedene kappa, unter folgenden Gesichtspunkten:
1. Konsistenz, Konsistenzstärke, Verträglichkeit mit grossen Kardinalzahleigenschaften von kappa,
2. Konsequenzen (derer gibt es eine Vielzahl),
3. Kombinierbarkeit der Prinzipien für verschiedene kappa gleichzeitig.
2. #### Stefan Bold (Bonn): Measure Analysis under the Axiom of Determinacy - Cardinals as Ultrapowers

April 17, 2007
Abstract: It is a result from Martin that ultrapowers on strong partition cardinals are cardinals. The Axiom of Determinacy gives us strong partition cardinals. So in this context we can talk about measure analysis, i.e., whether it is possible to describe cardinals as ultrapowers in a canonical way. So we want an assignment of measures to cardinals and vice versa that has the right properties.

In my talk I will give an overview of the apparatus neccessary for this task. The construction of the actual measure assignment is by induction, since the measures needed to analyse a cardinal are built from measures on cardinals below it.

I will present the first step of this inductive analysis of cardinals in more detail. And last but not least there are some nice applications that arise from the measure analysis.

3. #### Arthur Apter (CUNY): Large Cardinals with Few Measures

April 24, 2007
Abstract: I will discuss joint work with James Cummings and Joel Hamkins in which, starting from the consistency of one measurable cardinal, we force and obtain a model where the least measurable cardinal $\kappa$ carries exactly $\kappa^+$ many normal measures. This answers a question of Stewart Baldwin. The methods generalize to higher cardinals, showing that the number of $\lambda$ strong compactness or $\lambda$ supercompactness measures on $P_\kappa(\lambda)$ can be exactly $\lambda^+$, if $\lambda > \kappa$ is a regular cardinal. I will conclude with a list of open questions. The proofs use a new method due to James Cummings, along with Joel Hamkins' gap forcing techniques.
4. #### Arthur Apter (CUNY): A model in which every cardinal is almost Ramsey

May 8, 2007
Abstract: I will discuss forcing the existence of a choiceless model of ZF in which every (well-ordered) cardinal is almost Ramsey.
5. #### Informal Meeting of Set Theorists (IMST 2007)

Thursday May 17 (Himmelfahrt), 2007, all day
Talks by Jesse Alama (Stanford), Peter Koepke (Bonn), Benjamin Seyfferth (Bonn), Sergei Tupailo (Stanford) and Jindra Zapletal (University of Florida). For more details click here.
6. #### Joel Hamkins (CUNY): The Ground Axiom and V not= HOD

June 12, 2007
Abstract: The Ground Axiom asserts that the universe is not a nontrivial set-forcing extension of any inner model. Despite the apparent second-order nature of this assertion, Reitz proved that it is actually first-order expressible in set theory. The previously known models of the Ground Axiom all satisfy strong forms of V=HOD. Nevertheless, in recent joint work between myself, Reitz and Woodin, we have proved that the Ground Axiom is relatively consistent with V not= HOD. In fact, every model of ZFC has a class forcing extension that is a model of ZFC+GA+V not= HOD. The method is robust, and it accommodates large cardinals.
7. #### Russell Miller (CUNY): Locally Computable Structures

Thursday June 14, 2007, 16:30, SR A
Abstract: Computable model theory has always restricted itself to the study of countable structures. The natural numbers form the standard domain and range for partial computable functions, due to the finiteness of the programs and computations in the Turing model. Other models of computability do exist: infinite-time Turing machines and register machines (see [3] and [4]), the bitmap model for functions on Cantor space ([2]), and the Blum-Shub-Smale model ([1]), among others. However, by far the most widely accepted and used model of computation is the finite-time Turing machine, and almost all of computable model theory has been based on this model. Hence the restriction to countable structures. In this talk we continue to use the standard model of computation, but in such a way as to allow consideration of uncountable structures. We no longer aspire to describe a structure S globally. Instead, we content ourselves with a local description: a list of the finitely-generated substructures of S, up to isomorphism. Initially we require that these substructures should be computable, and that the list of them be given effectively. If this can be done, then S is said to be locally computable. (Such an S must have only countably many finitely-generated substructures, up to isomorphism.) Then we consider more sophisticated effective descriptions of S, by naming embeddings among the substructures which reflect the inclusions in S. We will prove some basic theorems about locally computable structures, but since the notion of local computability is extremely new, most of this talk will be devoted to definitions and examples. If time permits, we may offer some conjectures about how these ideas could interact with ordinal computability.
References: [1] L. Blum, M. Shub, & S. Smale; On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions, and universal machines, Bulletin of the AMS 21 (1989), 1-46. [2] M. Braverman & S. Cook; Computing over the Reals: Foundations for Scientific Computing, Notices of the AMS 53 (2006) 3, 318-329. [3] J.D. Hamkins and A. Lewis: Infinite time Turing machines, Journal of Symbolic Logic 65 (2000) 2, 567--604. [4] P. Koepke; Ordinals, computations, and models of set theory, preliminary version, January 2006.
8. #### Martin Koerwien (Paris): Die Komplexität der Isomorphierelation für abzählbare Modelle omega-stabiler Theorien.

June 26, 2007
9. #### Stefan Bold (Bonn): Measure Analysis under the Axiom of Determinacy - Cardinals as Ultrapowers, Part II

July 3, 2007
Last changed June 28, 2007