The $\hat{I}º$-Strongly Proper Forcing Axiom

Abstract

We study methods to obtain the consistency of forcing axioms, and particularly higher forcing axioms. We first force over a model with a supercompact cardinal $\hat{I}¸>\hat{I}º$ to get the consistency of the forcing axiom for $\hat{I}º$-strongly proper forcing notions which are also $\hat{I}º$-lattice, and then eliminate the need for large cardinals. The proof goes through a natural reflection property for $\hat{I}º$-strongly proper forcings. We also produce a model of this forcing axiom with $2^{\hat{I}}º$ arbitrarily large, and prove the inconsistency of certain natural strengthenings of the axiom.

14 pages