## Oberseminar mathematische Logik

### Organizers

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

### Time and location

Monday 16.00-18.00 in room 0.011, 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

- 20 April Peter Holy (Bonn): Condensation does not imply Square.

Abstract: All the known arguments to verify square principles to hold in L rely on some sort of fine structural machinery. It is generally believed that this is in fact necessary. We support this belief by showing that (a certain form of) Condensation does not imply square to hold. The large cardinal assumption for this proof will be a 2-Mahlo cardinal. This is joint work with Philip Welch and Liuzhen Wu. - 27 April Philipp Lücke (Bonn): Chain conditions, layered partial orders and weak compactness.

Abstract: Motivated by a conjecture of Todorcevic, we study strengthenings of the κ-chain conditions that are equivalent to the κ-chain condition in the case where κ is a weakly compact cardinal. We then use such properties to provide new characterizations of weakly compact cardinals. This is joint work in progress with Sean D. Cox (VCU Richmond). - 04 May Regula Krapf (Bonn): Π^1_1-Determinacy and topological regularity in second order arithmetic.

Abstract: It has been shown by Koepke/Moellerfeld/Dimitriou that within a model of second order arithmetic (SOA) with the Π^1_1-Perfect Subset Property one can construct an inner model of ZFC and conversely, by collapsing a model of ZFC one obtains a model of SOA+projective topological regularity. We will generalize this to compare models of SOA+Π^1_1-Determinacy+Π^1_2-Perfect Subset Property with models of ZFC where all sharps of sets of ordinals exist. - 11 May No seminar.

- 18 May Radek Honzik (Prague/Vienna): Satisfaction in outer models.

Abstract: We will study a generalized notion of satisfaction in which the collection of test structures is restricted to outer models of a given transitive set model M of ZFC. We will show that it is consistent from an inaccessible cardinal that there is M which can define in a lightface way satisfaction in its outer models (we say that M defines its outer model theory). The proof uses Barwise's results on infinitary logic L_{∞,ω}and a non-monotonic Easton-type iteration which manipulates the continuum function on regular cardinals in M and which is longer (in terms of ordinal type) than the number of ordinals in M. The result complements an unpublished result of Mac Stanley who showed that if M contains many Ramsey cardinal then it defines its outer model theory. The work is joint with Sy Friedman. - 25 May No seminar.

- 01 June Ana Njegomir (Bonn): Forcing with finite sequences of models of two types.

Abstract: We will introduce an approach to forcing with finite sequences of models of two types and show that this forcing is strongly proper for certain models. If the time allows us we will show some of its properties. Forcing with finite sequences of models as side conditions has many applications, one of them is the new proof of the consistency of PFA using finite support iteration. This was all introduced in a paper by Itay Neeman. - 08 June Philip Welch (Bristol): GRP: Global Reflection Principles.

Abstract: Reflection principles as commonly understood in set theory, are principles that regard the universe V of sets as somehow ineffable, and anything we say about V, in whatever language or logic must reflect down to some initial segment. However all such principles are *intra-constructible *that is they produce strong axiom of infinity that are in reality rather weak: they are consistent with V=L. As is well known, work of many other set theorists use the assumption of a proper class of Woodin cardinals in varying contexts. In this talk we sketch some history of Reflection Principles and consider *Global Reflection Principles *that take a naive Cantorian view of sets and classes and produces *extra-constrictible principles. *GRP can provide a proper class of measurable Woodin cardinals. - 15 June Carolin Antos-Kuby (Vienna): Hyperclass-Forcing in Morse-Kelley.

- 22 June Giorgio Laguzzi (Freiburg): Amoebas for uncountable cardinals.

Abstract: Amoeba forcings are rather important tools in set theory of the real numbers, since they can be used to get consistency results concerning regularity properties and cardinal invariants. We will investigate these notions in the framework of the generalized Cantor and Baire spaces. We will see how the properties of these forcings are strongly influenced by the cardinals we deal with. - 29 June No seminar.

- 06 July Sandra Uhlenbrock (Münster): Mice with finitely many Woodin cardinals from optimal determinacy hypotheses

Abstract: Mice are countable sufficiently iterable models of set theory. Itay Neeman has shown that the existence of such mice with finitely many Woodin cardinals implies that projective determinacy holds. In fact he proved that the existence and ω_{1}-iterability of M^{#}_{n}(x) for all reals x implies that boldface Π^{1}_{n+1}-determinacy holds. We prove the converse of this result, that means boldface Π^{1}_{n+1}-determinacy implies that M^{#}_{n}(x) exists and is ω_{1}-iterable for all reals x. This level-wise connection between mice and projective determinacy is an old so far unpublished result by W. Hugh Woodin. As a consequence we can obtain the determinacy transfer theorem for all levels n. These results connect the areas of inner model theory and descriptive set theory, so we will give an overview of the relevant topics in both fields and briefly sketch a proof of the result mentioned above. The first goal is to show how to derive a model of set theory with Woodin cardinals from a determinacy hypothesis. The second goal is to prove that there is such a model which is iterable. For this part the odd and even levels of the projective hierarchy are treated differently. This is joint work with Ralf Schindler and W. Hugh Woodin. - 08 July Frank Stephan (Singapore): Automatic Structures - recent results and open questions (Wednesday, 10.30-12.00 in room 0.003).

Abstract: Automatic structures are a way to represent algebraic structures using finite automata; all the algebraic operations and relations have to be recognised / verified by finite automata which read the inputs and outputs with the same speed (one symbol per cycle). The talk gives an overview on what structures can be represented this way and which questions are left open in the field. A paper is available. - 09 July Assaf Rinot (Ramat-Gan): Chain conditions of products (Thursday, 16.00-18.00 in room 1.007).

Abstract: We shall survey the history of the study of the productivity of the κ-cc in partial orders, topological spaces, and Boolean algebras. We shall address a conjecture that tries to characterize such a productivity in Ramsey-type language. For this, a new oscillation function for successor cardinals, and a new characteristic function for walks on ordinals will be proposed and investigated. - 10 September Asger Törnquist (Copenhagen): TBA (Thursday, 16.00-18.00 in room 1.008).