## Philipp Lücke: Squares, chain conditions, and products

Monday, November 13, 2017, 16.30
Seminar room 0.008, Mathematical Institute, University of Bonn

Speaker: Philipp Lücke (Universität Bonn)

Title: Squares, chain conditions, and products

Abstract:

With the help of square principles, we obtain results concerning the consistency strength of several statements about strong chain conditions and their productivity. In particular, we show that if the κ-Knaster property is countably productive for some uncountable regular cardinal κ, then κ is weakly compact in L. The proof of this result relies on a new construction that shows that Todorcevic’s principle □(κ) implies an indexed version of the principle □(κ,λ). This is joint work with Chris Lambie-Hanson (Bar-Ilan).

## Peter Holy: The exact strength of the class forcing theorem

Monday, October 23, 2017, 16.30
Seminar room 0.008, Mathematical Institute, University of Bonn

Speaker: Peter Holy (Universität Bonn)

Title: The exact strength of the class forcing theorem

Abstract:

We consider second order set theories, that have as objects both sets and classes, and the role of the class forcing theorem, that is the forcing theorem for all notions of class forcing, within this range of theories. While Kelley-Morse class theory (KM) proves the class forcing theorem, its failure is consistent with the axioms of Gödel-Bernays set theory (GBC). We show that the class forcing theorem is equivalent, over GBC, to the principle of elementary transfinite (class) recursions of length Ord, and to the existence of various kinds of truth predicates. This is joint work with Victoria Gitman, Joel Hamkins, Philipp Schlicht and Kameryn Williams.

## Stefan Hoffelner: NS saturated and Delta_1-definable

Monday, June 19, 2017, 16.30
Seminar room 0.011, Mathematical Institute, University of Bonn

Speaker: Stefan Hoffelner (University of Vienna)

Title: NS saturated and Delta_1-definable

Abstract:

Questions which investigate the interplay of the saturation of the nonstationary ideal on $omega_1$, NS, and definability properties of the surrounding universe can yield surprising and deep results. Woodins theorem that in a model with a measurable cardinal where NS is saturated, CH must definably fail is the paradigmatic example. It is another remarkable theorem of H. Woodin that given $omega$-many Woodin cardinals there is a model in which NS is saturated and $omega_1$-dense, which in particular implies that NS is (boldface) $Delta_1$-definable. S.D. Friedman and L. Wu asked whether the large cardinal assumption can be lowered while keeping NS $Delta_1$-definable and saturated. In this talk I will outline a proof that this is indeed the case: given the existence of $M_1^{#}$, there is a model of ZFC in which the nonstationary ideal on $omega_1$ is saturated and $Delta_1$-definable with parameter $K_{omega_2^K}$ (note that $omega_2^K$ is of size $aleph_1$ in that model). In the course of the proof I will present a new coding technique which seems to be quite suitable to obtain definability results in the presence of iterated forcing constructions over inner models for large cardinals.

## Yizheng Zhu: Iterates of M_1

Monday, June 12, 2017, 16.30
Seminar room 0.011, Mathematical Institute, University of Bonn

Speaker: Yizheng Zhu (University of Münster)

Title: Iterates of M_1

Abstract:

## Ralf Schindler: A Hamel basis for the reals without choice

09 January: 16.30
seminar room 0.008
Mathematisches Institut
Universität Bonn
Endenicher Allee 60

Ralf Schindler (Münster) – A Hamel basis for the reals without choice

The Cohen-Halpern-Levy model N has an infinite set of reals without a countable subset. Answering a question of D. Pincus and K. Prikry from 1975, we show that there is a Hamel basis in N. This is joint work with Liuzhen Wu and Liang Yu, inspired by earlier joint work with Mariam Beriashvili. DC fails in N, and it remains open if in the base theory ZF+DC, the existence of a Hamel basis implies that the reals can be wellordered.

## David Schrittesser: Maximal discrete sets with large continuum

25.01.2016
Seminar room 0.011, Mathematical Institute, University of Bonn

Speaker: David Schrittesser (University of Copenhagen)

Title: Maximal discrete sets with large continuum

Abstract. In recent joint work with Asger Törnquist, we showed how to construct definable maximal discrete sets in forcing extensions of L, in particular in the Sacks and Miller extension. In particular, the existence of such sets is consistent with V ≠ L. In this talk I shall show the stronger result that the existence of definable discrete sets is consistent with large continuum. In the process, I show an interesting generalization of Galvin’s theorem. In particular, this applies to the example of maximal orthogonal families of measures (mofs). One might hope for a simpler way of constructing a mof in a model with large continuum: to find an indestructible such family in L. While such an approach is possible e.g. for maximal cofinitary groups, this is impossible for mofs.

## Asger Törnquist: Definable maximal orthogonal families and discrete sets in forcing extensions

Thursday, September 10, 2015, 16.30
Seminar room 1.008, Mathematical Institute, University of Bonn

Speaker: Asger Törnquist (Kopenhagen)

Title: Definable maximal orthogonal families and discrete sets in forcing extensions