Many forcing axioms for all regular uncountable cardinals

TL;DR

This paper explores the consistency of forcing axioms for all regular uncountable cardinals within a universe satisfying GCH, focusing on stationary sets and their properties, with implications for algebraic structures.

Contribution

It introduces a method to construct models satisfying GCH that support many forcing axioms related to stationary sets for all regular uncountable cardinals.

Findings

Successfully models GCH with numerous stationary set-based forcing axioms.

Demonstrates the consistency of these axioms across all regular uncountable cardinals.

Provides a framework for analyzing the interplay between set theory and algebraic properties.

Abstract

Our original aim was, in Abelian group theory to prove the consistency of: lambda is strong limit singular and for some properties of abelian groups which are relatives of being free, the compactness in singular fails. In fact this should work for R-modules, etc. As in earlier cases part of the work is analyzing how to move between the set theory and the algebra. Set theoretically we try to force a universe which satisfies G.C.H. and diamond holds for many stationary sets but, for every regular uncountable lambda, in some sense anything which "may" hold for some stationary set, does hold for some stationary set. More specifically we try to get a universe satisfying GCH such that e.g. for regular kappa < lambda there are pairs (S,B), S \subseteq S^\lambda_\kappa stationary, B \subseteq H (lambda), which satisfies some pregiven forcing axiom related to (S,B), (so (lambda\ S)-complete, i.e. "trivial outside S) but no more, i.e. slightly stronger versions fail. So set theoretically we try to get a universe satisfying G.C.H. but still satisfies "many", even for a maximal family in some sense, of forcing axioms of the form "for some stationary" while preserving GCH. As completion of the work lagged for a while, here we deal only with the set theory.