The $\kappa$-Strongly Proper Forcing Axiom

TL;DR

This paper develops methods to establish the consistency of the $ ext{ extkappa}$-Strongly Proper Forcing Axiom, including eliminating large cardinal assumptions and analyzing its strength and limitations.

Contribution

It introduces a new approach to prove the consistency of the $ ext{ extkappa}$-Strongly Proper Forcing Axiom without large cardinals and explores its implications.

Findings

Established the consistency of the forcing axiom for $ ext{ extkappa}$-strongly proper forcings.

Produced models with arbitrarily large $2^ ext{ extkappa}$ under the axiom.

Proved the inconsistency of certain natural strengthenings of the 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 $\theta>\kappa$ to get the consistency of the forcing axiom for $\kappa$-strongly proper forcing notions which are also $\kappa$-lattice, and then eliminate the need for large cardinals. The proof goes through a natural reflection property for $\kappa$-strongly proper forcings. We also produce a model of this forcing axiom with $2^\kappa$ arbitrarily large, and prove the inconsistency of certain natural strengthenings of the axiom.