July 17
Kaethe Minden, Bard College at Simon's Rock
Maximality and Resurrection

The maximality principle (${\rm MP}$) is the assertion that any sentence which can be forced in such a way that after any further forcing the sentence remains true, must already be true. In modal terms, ${\rm MP}$ states that forceably necessary sentences are true. The resurrection axiom (${\rm RA}$) asserts that the ground model is as existentially closed in its forcing extensions as possible. In particular, ${\rm RA}$ relative to $H_{\mathfrak c}$ states that for every forcing $\mathbb Q$ there is a further forcing $\mathbb R$ such that $H_{\mathfrak c}^V \prec H_{\mathfrak c}^{V[G][H]}$, for $G*H \subseteq \mathbb Q *\dot{\mathbb R}$ generic.

It is reasonable to ask whether ${\rm MP}$ and ${\rm RA}$ can consistently both hold. I showed that indeed they can, and that ${\rm RA}+{\rm MP}$ is equiconsistent with a strongly uplifting fully reflecting cardinal, which is a combination of the large cardinals used to force the principles separately. In this talk I give a sketch of the equiconsistency result.