## Oberseminar mathematische Logik

### Organizers

- Prof. Dr. Peter Koepke
- Dr. Philipp Lücke
- Dr. Philipp Schlicht

### Time and location

Mondays 16.30-18.00 in room 1.007, Endenicher Allee 60.

The participants of the seminar are welcome for coffee and tea in the Plückerraum 1.012 at 16.00 before the talks.

### Talks

- 18 April: Asaf Karagila (Jerusalem) - The long shadow of a Cohen real: The Bristol model

The Bristol model gets its name from the workshop on the HOD conjecture that took place in Bristol in 2011. The model is related to the "AC Conjecture" which states that if the HOD conjecture is provable, the axiom of choice can be forced by a set forcing in the presence of sufficiently large cardinals. The model itself is not a counterexample, but it is a first step towards such a counterexample. - 25 April: Anne Fernengel (Bonn) - An Easton-like Theorem for Zermelo-Fraenkel Set Theory without Choice (Part I.)

In ZFC, Easton's Theorem states that for regular cardinals, the only constraints on the continuum function 2^{κ}are weak monotonicity and König's Lemma. For singular cardinals, however, the situation is a lot more involved, since the 2^{κ}-values for singular κ are strongly influenced by the behavior of the continuum function below. We show that in Zermelo-Fraenkel Set Theory without AC, the θ-function, a surjective substitute for the continuum function 2^{κ}, can take almost arbitrary values for all infinite cardinals. This choiceless version of Easton' Theorem is in sharp contrast to the situation in ZFC. - 02 May: Anne Fernengel (Bonn) - An Easton-like Theorem for Zermelo-Fraenkel Set Theory without Choice (Part II.)
- 09 May: Peter Holy (Bonn) - Small embedding characterizations for large cardinals, and internal large
cardinals

A small embedding characterization of a large cardinal is one where the critical point of the embedding(s) in question is mapped to the large cardinal it defines. Classical examples of small embedding characterizations are those for subcompactness and Magidors characterization of supercompactness. I will show that many other large cardinals, and in particular some of the smallest of large cardinals, possess small embedding characterizations. Moreover I will introduce some of the ideas behind our concept of internal large cardinal, that relate to small embedding characterizations in a way similar to how generic large cardinals relate to usual embedding characterizations of large cardinals (those where the large cardinal in question is characterized by being the critical point of the relevant embedding(s)).

This is ongoing joint work with Philipp Lücke. - 23 May: Philipp Schlicht (Bonn) - A generalization of the Baire property

Some simply definable subsets of generalized Baire spaces and Cantor spaces do not have the property of Baire, for instance the club filter and the non-stationary ideal. Therefore many applications of the property of Baire cannot be directly extended to the uncountable context. We define a natural analogue to the property of Baire for generalized Baire spaces. We study some of its consequences and show that it is consistent that this property holds for all subsets of generalized Baire spaces with sufficiently simple definitions. - 30 May: no seminar
- 03 June Friday 16:30-18:00, room 1.008: Michal Doucha (Franche-Comté) - Metric Scott analysis

The Scott analysis is an important tool of the classical infinitary logic developed already in 1950s and 1960s. It allows to assign to any countable structure a sentence in infinitary logic as a complete invariant, i. e. a sentence which describes the structure uniquely up to isomorphism. Moreover, it provides Borel approximations for the isomorphism relation. I will provide an introduction to the classical Scott analysis and then present our results which are generalization of the Scott analysis for metric structures and continuous infinitary logic. I will show some applications to the study of the isomorphism relation of metric structures, to the study of the Gromov-Hausdorff distance between Polish metric spaces and the Kadets distance between Banach spaces, and characterization of Polish CLI groups.

This is joint work with Itai Ben Yaacov, Andre Nies and Todor Tsankov. - 06 June: Philipp Lücke (Bonn) - Forcings that characterize large cardinals

The relative consistency of many combinatorial principles can be established by collapsing a large cardinal to be the successor a smaller regular cardinal. In many important cases, it turns out that the given large cardinal assumption was necessary, because the principle implies that the corresponding successor cardinal has the same large cardinal properties in some canonical inner model. We will consider the question whether certain collapse forcings characterize large cardinal properties through the validity of combinatorial principles in their forcing extensions, in the sense that the collapse forces the principle to hold if and only if the collapsed cardinal possess the corresponding large cardinal property in the ground model. It is easy to see that Levy collapses cannot characterize inaccessible cardinals. In contrast, the class of forcings that first add a Cohen real and then Levy collapse a cardinal to be ω_{2}characterizes many important types of large cardinals, like inaccessible, Mahlo, weakly compact, indescribable, measurable, supercompact and huge cardinals.

This is joint work in progress with Peter Holy (Bonn). - 10 June Friday 16:30-18:00, room 1.007: Julian Schlöder (Amsterdam) - Weak Rejection and Classical Negation

The central question of metalogic is about the meaning of the connectives. A particularly contentious case is the negation operator: should we admit double negation elimination and, if so, on what grounds? I will defend the rejectivist position, i.e. that negation can be explained through the dialogical act of rejection (or denial). The received view has it that rejection is less specific than assertion and hence cannot or should not be included as a primitive notion in logic. In particular, rejections that are less informative than a negative assertion are regarded as too unspecific for logical inference. The latter claim seems to be overly pessimistic. I will present a logic of weak rejection, motivated by principles of coherence in dialogue, that is sound and complete for two natural semantics. The logic derives classical propositional logic as valid on its asserted fragment, i.e. on the fragment that is about truth.

This is joint work with Luca Incurvati. - 13 June: no seminar
- 20 June: no seminar
- 27 June: Aleksandra Kwiatkowska (Bonn) - Groups of measurable functions

We study properties of groups L^{0}([0,1],μ,G) of measurable functions defined on the standard Lebesgue space ([0,1],μ) with values in a Polish group G. After discussing what is known about the structure and dynamics of such groups, I will present a few new results. In particular, I will give a sufficient condition for the existence of a cyclically dense conjugacy class in a group L^{0}([0,1],μ,G), which I will illustrate by giving several examples.

This is joint work with Maciej Malicki. - 05 July Tuesday 17.00-18.00, room 1.007: Frank Stephan (Singapore) - Finitely generated semiautomatic groups

Abstract: The present work shows that Cayley automatic groups are semiautomatic and exhibits some further constructions of semiautomatic groups. Furthermore, the present work establishes that every finitely generated group of nilpotency class 3 is semiautomatic.

This is joint work with Sanjay Jain and Bakhadyr Khoussainov.

Slides - 11 July: Raphaël Carroy (Turin) - When embeddings meet epimorphisms

We call strongly surjective a linear order that surjects order-preservingly onto each of its sub-orders. We will see that countable strongly surjective orders are complete among the sets that are the union of an analytic and a coanalytic set. If time allows we will also see that under PFA there are uncountable strongly surjective orders.