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.

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