The $κ$-Strongly Proper Forcing Axiom
David Asperó, Sean Cox, Asaf Karagila, Christoph Weiss
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 $θ>κ$ to get the consistency of the forcing axiom for $κ$-strongly proper forcing notions which are also $κ$-lattice, and then eliminate the need for large cardinals. The proof goes through a natural reflection property for $κ$-strongly proper forcings. We also produce a model of this forcing axiom with $2^κ$ arbitrarily large, and prove the inconsistency of certain natural strengthenings of the axiom.