Archives of: Bristol Logic Seminar

David Aspero: Generic absoluteness for Chang models

Tuesday, March 21, 2017, 15.00
Howard House 4th Floor Seminar Room

Speaker: David Aspero (University of East Anglia)

Title: Generic absoluteness for Chang models

Abstract:

The main focus of the talk will be on extensions of Woodin’s classical result that, in the presence of a proper class of Woodin cardinals, C_omega^V and C_omega^{V^P} are elementarily equivalent for every set—forcing P (where C_kappa denotes the kappa—Chang model).

1. In the first part of the talk I will present joint work with Asaf Karagila in which we derive generic absoluteness for C_omega over the base theory ZF+DC.

2. Matteo Viale has defined a strengthening MM^{+++} of Martin’s Maximum which, in the presence of a proper class of sufficiently strong large cardinals, completely decides the theory of C_{omega_1} modulo forcing in the class Gamma of set—forcing notions preserving stationary subsets of omega_1, i.e., if MM^{+++} holds, P is in Gamma, and P forces MM^{+++}, then C_{omega_1}^V and C_{omega_1}^{V^P} are elementarily equivalent. MM^{+++} is the first example of a “category forcing axiom.”

In the second part of the talk I will present some recent joint work with Viale in which we extend his machinery to deal with other classes Gamma of forcing notions, thereby proving the existence of several mutually incompatible category forcing axioms, each one of which is complete for the theory of C_{omega_1}, in the appropriate sense, modulo forcing in Gamma.

Peter Holy: A Hierarchy of Ramsey cardinals

Tuesday, March 7, 2017, 15.00
Howard House 4th Floor Seminar Room

Speaker: Peter Holy (Hausdorff Centre, University of Bonn)

Title: A Hierarchy of Ramsey cardinals

Abstract:

I will introduce a hierarchy of large cardinal notions in the area of Ramsey cardinals, that in particular are closely related to Victoria Gitman’s Ramsey-like cardinals. I will show this hierarchy to be a proper hierarchy, and the cardinals in this hierarchy to have a range of equivalent characterizations, using either infinite games, elementary embeddings or filters. I will try to argue that the cardinals kappa that top our hierarchy, which are what we call the kappa-Ramsey cardinals, may be seen as a more natural (and slightly stronger) version of Gitman’s super Ramsey cardinals. This is joint work with Philipp Schlicht.

Monika Seisenberger: Programs from constructive and classical proofs

Tuesday, February 28, 2017, 15.00
Howard House 4th Floor Seminar Room

Speaker: Monika Seisenberger (Swansea University)

Title: Programs from constructive and classical proofs

Abstract:

Program extraction from formal proofs is a powerful proof theoretic technique based on realisability to obtain provably correct programs. In this talk we give a short overview on the state-of-the-art, give several examples for program extraction from constructive proofs in different areas (Well-quasiorders, Sat Solving, Parsing), and also show how classical proofs which are often more concise and elegant compared to a constructive counterpart can lead to interesting programs. The case studies are carried out in the interactive theorem prover Minlog.

Philipp Lücke: Sigma_1-partition properties

Tuesday, February 14, 2017, 15.00
Howard House 4th Floor Seminar Room

Speaker: Philipp Lücke (Hausdorff Centre, University of Bonn)

Title: Sigma_1-partition properties

Abstract:

We consider colourings of the set of pairs of countable ordinals with two colours that are definable by Sigma_1-formulas that only use the first uncountable cardinal omega_1 and real numbers as parameters. We present results showing that the existence of a measurable cardinal above a Woodin cardinal implies that uncountable homogeneous sets exist for all such colourings. In contrast, a failure of this partition property is compatible with the existence of a single Woodin cardinal. Finally, we show that similar definable partition properties can hold for large cardinals that are not weakly compact; e.g. stationary limits of omega_1-iterable cardinals.

Charles Morgan: Forcing with small working parts

Tuesday, January 31, 2017, 15.00
Howard House 4th Floor Seminar Room

Speaker: Charles Morgan (University of Bristol)

Title: Forcing with small working parts

Abstract:

I will discuss set theoretic construction with ‘small working parts’, from its roots in the 1970s to current developments in adding combinatorial structures on $\omega_2$ and proving various forcing axioms consistent. A substantial part of the focus will the on the allied technique of using models as ‘side conditions’.

Philip Welch: The Ramified Analytical Hierarchy and Strong Logics

Tuesday, January 17, 2017, 15.00
Howard House 4th Floor Seminar Room

Speaker: Philip Welch (University of Bristol)

Title: The Ramified Analytical Hierarchy and Strong Logics

Abstract:

The ramified analytical hierarchy defined by Kleene builds up a hierarchy of models of subsystems of analysis in a second order definable manner.

We address a question of Kennedy as to what can be done using strong logics to re-define the stages of Kleene’s hierarchy, in the spirit of “Inner Models from Extended Logics” of Kennedy, Magidor, & Väänänen. In this paper they followed a suggestion of Gödel that the definability function used to build the levels of the constructible hierarchy be modified to make use of stronger logics. The resultant hierarchy might, or might not, then be L itself. We show that by changing the logic in the ramified analytical hierarchy allows one to construct, eg., the minimal ‘correct’ model of analysis.