## Oberseminar Logik - Advanced logic seminar

Advanced talks on mathematical logic by guests and members of the logic group.

*Master module S4A3, 6 credit points, workload 180h*

## Organized by

## Prof. Dr. Peter Koepke

### Time and location

Tuesdays 16:30-18:00 at seminar room A, Beringstrasse 4.

The members of the seminar are welcome for coffee and tea at
Peter Koepke's office, Room 44, Beringstrasse 4
from 16:00-16:30 before the talks.

### Programme

**Thursday 17 April**Arthur Apter (CUNY)*"The least uncountable regular cardinal can be the least measurable cardinal"*-

I will discuss how to force and construct choiceless models of ZF in which the least regular uncountable cardinal κ is the least measurable cardinal. To avoid trivialities, it will always be the case that κ > ω_{1}.

*[ Note the change of day and place: Thursday at the Hausdorff room, ground floor of Beringstrasse 3 (16:30-18:30) ]*

**Monday 21 April**Ioanna Dimitriou (Bonn)*"A modern approach to Gitik's `All uncountable cardinals can be singular'"*

*[Note the change of day and time: Monday 10:00-12:00 (SR C)]*

**Tuesday 29 April**"Julia Robinson and Hilbert's Tenth Problem" documentary film

A one-hour biographical documentary,*Julia Robinson and Hilbert's Tenth Problem*tells the story of an important American mathematician against a background of mathematical ideas. The film will focus on the individual and collaborative role of Julia Robinson (1919-1985), a past-president of the American Mathematical Society (AMS), and on the friendship and collaboration that developed between her and Yuri Matiyasevich after he put the last necessary piece of the solution to H10 into place. Telling Robinson's story on screen will be her sister and biographer, Constance Reid, who is the author of several noted mathematical biographies, including one on David Hilbert.

[Note the change of place: at the Hausdorff room (16:30-18:30)]

**Tuesday 27 May**Daniel Herden (Jerusalem)*"On the construction of generalized E-rings"*-
Since their introduction by Laszlo Fuchs in 1958 (generalized) E-rings share a long and vivid history of research results and surprising applications. Here a ring

*R*is called a generalized E-ring if it is isomorphic to the ring End*R*^{+}of group endomorphisms of its additive structure*R*^{+}.

The topic of our double talk will be the proof on the existence of non-commutative generalized E-rings. We will focus especially on the model theoretic and set theoretic aspects of the construction. **Monday 2 June**Thilo Weinert (Münster)*"BPFA and BAAFA, their equiconsistency and their nonequivalence"*

In recent years set theorists have increasingly worked on Bounded Forcing Axioms. Interest in them might be due to the possibility of reformulating them as principles of generic absoluteness, but also to the fact that many of them decide the size of the continuum.

The weakest Forcing Axiom presently shown to do so, the Bounded Proper Forcing Axiom(BPFA), is a rather modest statement in terms of consistency strength. It is equiconsistent to the existence of a reflecting cardinal—stationarily many of which lie below any Mahlo cardinal. Considering the class of forcing notions satisfying Axiom A—which is a proper subclass of all proper notions of forcing—one can formulate the Bounded Axiom A Forcing Axiom(BAAFA). It turns out that it is indeed a proper weakening of BPFA, i.e. there is a forcing extension in which BAAFA holds, but BPFA fails. Indeed the consistency strengths of BPFA, BAAFA and BAAFA∧¬BPFA are all the same.

[Note the change of day: Monday]**Tuesday 10 June**Rüdiger Göbel (Duisburg - Essen)*"On the construction of generalized E-rings, part 2 "*

- Since their introduction by Laszlo Fuchs in 1958 (generalized) E-rings share a long and vivid history of research results and surprising applications. Here a ring
*R*is called a generalized E-ring if it is isomorphic to the ring End*R*^{+}of group endomorphisms of its additive structure*R*^{+}.

The topic of our double talk will be the proof on the existence of non-commutative generalized E-rings. We will focus especially on the model theoretic and set theoretic aspects of the construction.

**CiE 2008 Athens - Contributions by the Bonn Logic Group****Tuesday 24 June**Andy Lewis (Leeds)*"Constructing minimal degrees"*-
**Tuesday 1 July**Philipp Schlicht (Münster)*"Projective absoluteness and thin equivalence relations"*

Σ_{n}^{1}absoluteness for a forcing means that the same Σ_{n}^{1}facts about reals in the ground model are true in the ground model and in the generic extension. A related property is that there are no new equivalence classes of thin**Π**_{n-1}^{1}and**Σ**_{n-1}^{1}equivalence relations (equivalence relations with no perfect set of pairwise inequivalent reals) in the generic extension. We discuss how these properties are connected and prove both simultaneously for reasonable forcing, assuming the existence of M_{n-3}^{#}(X) for all transitive sets X of the same size as the forcing. The proof is based on an idea of Foreman and Magidor. We further present a version of this result for**Σ**_{2}^{1}c.c.c. forcing in the context of projective determinacy.-
**Tuesday 8 July**Peter Koepke (Bonn)*"The consistency strength of the negation of the singular cardinal hypothesis without the axiom of choice"*

- We show - joint with Arthur Apter - that the consistency strengths of the theories ZF + ¬SCH and ZFC + ''there exists a measurable cardinal'' are equal, where ¬SCH is interpreted as: there is a singular cardinal κ such that ∀ν∈[ω,κ)
*P*(ν) ~ ν^{+}and there exists a*surjection*from*P*_{≤cof(κ)}(κ) onto κ^{++}. The proofs involve a standard application of the Dodd-Jensen core model theory and choiceless submodels of generic extensions by parallel Prikry forcing for a sequence of κ^{++}distinct normal measures on κ. -
**Thursday 17 July**An afternoon in ordinal computability with the talks that our group contributed to this year's CiE in Athens*(at Peter Koepke's office, 44/Be4)*

14:15-15:00 Koepke: ITRM

15:00-15:45 Clockable Ordinals

15:45-16:15 Pause

16:15-17:00 Lost Melody

17:00-17:45 Irrgang/Seyfferth

17:45-18:00 Kurze Pause

18:00- .. Athen-Bilder, Brezeln, Feta-Käse, Bier, Wein, ...

*[Note the change of day and place: Thursday in Peter Koepke's office]*