The $κ$-Strongly Proper Forcing Axiom

David Asperó, Asaf Karagila

Published 2019-12-04Version 1

We study methods with which we can obtain the consistency of forcing axioms, and particularly higher forcing axioms. We first prove that the consistency of a supercompact cardinal $\theta>\kappa$ implies the consistency of a forcing axiom for $\kappa$-strongly proper forcing notions which are also $\kappa$-lattice, and then eliminate the need for the supercompact cardinal. The proof goes through a natural reflection property for $\kappa$-strongly proper forcings and through the fact that every $\kappa$-sequence of ordinals added by a $\kappa$-lattice and $\kappa$-strongly proper forcing is in a $\kappa$-Cohen extension.