Archives of: Bonn Logic Seminar

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:

Assume Delta^1_3-determinacy. Let L_{kappa_3}[T_2] be the admissible closure of the Martin-Solovay tree and let M_{1,infty} be the direct limit of$M_1 via countable trees. We show that L_{kappa_3}[T_2]cap V_{u_{omega}} = M_{1,infty} | u_{omega}.

Ana Njegomir: A forcing characterization of lambda-ineffable cardinals

Monday, May 29, 2017, 16.30
Seminar room 0.011, Mathematical Institute, University of Bonn

Speaker: Ana Njegomir (Universität Bonn)

Title: A forcing characterization of lambda-ineffable cardinals

Andrey Morozov: Infinite time Blum-Shub-Smale machines for computability in analysis

Monday, May 15, 2017, 16.30
Seminar room 0.011, Mathematical Institute, University of Bonn

Speaker: Andrey Morozov (Sobolev Institute of mathematics, Novosibirsk)

Title: Infinite time Blum-Shub-Smale machines for computability in analysis

Chris Lambie-Hanson: Constructions from square and diamond, with an application to super-Souslin trees

Monday, May 8, 2017, 16.30
Seminar room 0.011, Mathematical Institute, University of Bonn

Speaker: Chris Lambie-Hanson (Bar-Ilan)

Title: Constructions from square and diamond, with an application to super-Souslin trees

Abstract. In 1982, Shelah and Stanley proved that, if $\kappa$ is a regular, infinite cardinal, $2^\kappa = \kappa^+$, and there is a $(\kappa^+, 1)$-morass, then there is a $\kappa^{++}$-super-Souslin tree, which is a type of normal $\kappa^{++}$-tree that necessarily has a $\kappa^{++}$-Souslin subtree and continues to do so in any outer model in which $\kappa^{++}$ is preserved and no new subsets of $\kappa$ are present. This result establishes a lower bound of an inaccessible cardinal for the consistency strength of the conjunction of $2^\kappa = \kappa^+$ and Souslin’s Hypothesis at $\kappa^{++}$. In this talk, we will present a method for constructing objects of size $\lambda^+$ from $\square_\lambda + \diamondsuit_\lambda$, where $\lambda$ is a regular, uncountable cardinal. As an application, we will use $\square_{\kappa^+} + \diamondsuit_{\kappa^+}$ to construct a $\kappa^{++}$-super-Souslin tree. For uncountable $\kappa$, this increases Shelah and Stanley’s lower bound from an inaccessible cardinal to a Mahlo cardinal. This is joint work with Assaf Rinot.

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

Assaf Rinot: Chain conditions of products

Thursday, July 9, 2015, 16.00
Seminar room 0.011, Mathematical Institute, University of Bonn

Speaker: Assaf Rinot (Bar-Ilan University)

Title: Chain conditions of products

Abstract: We shall survey the history of the study of the productivity of the k-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.

Giorgio Laguzzi: Roslanowski and Spinas dichotomies

Monday, July 14, 2014, 16.30
Seminar room 0.011, Mathematical Institute, University of Bonn

Speaker: Giorgio Laguzzi (Universität Hamburg)

Title: Roslanowski and Spinas dichotomies