# Matteo Viale: Well Behaving Category Forcings

Friday Set Theory Seminar (HUJI)

We shall meet next Friday (July 10th) in the Hebrew University math department building in Room 110, at 10 am.

Speaker: Matteo Viale (UNITO)

Title: Well Behaving Category Forcings

Abstract: We isolate and study certain nice properties for classes of forcings  : that of being $\kappa$-iterable (property A), that of being $\Pi^1_1$-persistent at $\kappa$ (property B), and that of having the
strong freezeability property (property C)..

We show that for classes of forcings having properties A,B,C there is a forcing axiom CFA($\Gamma$) (strentghtening the forcing axioms for $\kappa$-many dense sets for posets in $\Gamma$) with the property that no forcing in  preserving CFA($\Gamma$ ) can change the theory of the Chang model $L[Ord{}^{\le\kappa}$] with parameters in $H_{\kappa^+}$.
We also nd nine distinct classes $\Gamma$ which satisfy the above properties A,B,C for  = $\omega_1$.
For one of these classes  $\Gamma$, CFA($\Gamma$ ) implies CH (to be double checked). The other classes $\Gamma$ strengthen known forcing axioms such as MM, PFA and variations theoreof.
Moreover we can easily separate two distinct theories of the form ZFC + CFA($\Gamma_j$) using distinct $\Pi_2$-properties of the corresponding $H_{\omega_2}$.
For $\kappa>\omega_1$, we cannot give any example of a $\Gamma$ which can satisfy properties A,B,C, but any such $\Gamma$ will provide an interesting example of a forcing axiom for $\kappa$-many dense sets…..
This is a joint project with David Aspero.