Many forcing axioms for all regular uncountable cardinals

TL;DR

This paper explores the development of strong forcing axioms applicable to all regular uncountable cardinals under GCH, aiming to understand universes with highly controlled set-theoretic properties.

Contribution

It introduces a framework for forcing axioms that extend beyond traditional approaches, considering families of forcing notions dependent on stationary sets for all regular uncountable cardinals.

Findings

Framework for higher-order forcing axioms across all regular uncountable cardinals

Potential applications to Abelian group theory are identified

Establishment of conditions under GCH for these axioms

Abstract

A central theme in set theory is to find universes with extreme, well-understood behaviour. The case we are interested in is assuming GCH and has a strong forcing axiom of higher order than usual. Instead of "for every suitable forcing notion for~$\lambda$" we shall say "for every such family of forcing notions, depending on stationary $S\subseteq \lambda$, for some such stationary set we have\dots". Such notions of forcing are important for Abelian group theory, but this application is delayed for a sequel.