# Logic & Set Theory Seminar: Generic absoluteness for Chang models

21 March 2017, 3.00 PM - 21 March 2017, 4.30 PM

David Aspero UEA

Howard House 4th Floor Seminar Room

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, $\mathcal C_\omega^V$ and $\mathcal C_\omega^{V^P}$ are elementally equivalent for every set—forcing $P$ (where $\mathcal C_\kappa$ 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 $\mathcal C_\omega$ 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 $\mahcal C_{\omega_1}$ modulo forcing in the class $\Gamma$ of set—forcing notions preserving stationary subsets of $\omega_1$, i.e., if $MM^{+++}$ holds, $P\in \Gamma$, and $P$ forces $MM^{+++}$, then $\mathcal C_{\omega_1}^V$ and $\mathcal C_{\omega_1}^{V^P}$  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 $\mathcal C_{\omega_1}$, in the appropriate sense, modulo forcing in $\Gamma$.

Philip Welch