Generalizations of Martin's Axiom, weak square, weak Chang's Conjecture, and a forcing axiom failure

TL;DR

This paper explores the implications of certain forcing axioms on combinatorial principles, demonstrating inconsistencies and deriving weak Chang's Conjecture through logical implications.

Contribution

It establishes new connections between forcing axioms and combinatorial principles, showing that some axioms imply principles like _{ ext{omega}_1, ext{omega}_1} and weak Chang's Conjecture, and proves the inconsistency of specific forcing axioms.

Findings

MA^{1.5}_{\u2206_2}( ext{stratified}) implies _{ ext{omega}_1, ext{omega}_1}

MM_{\u2206_2}(\u2206_2 ext{-c.c.}) is inconsistent

Weak Chang's Conjecture follows from MA^{1.5}_{\u2206_2}( ext{stratified})

Abstract

We prove that the forcing axiom $MA^{1.5}_{\aleph_2}(\mbox{stratified})$ implies $\Box_{\omega_1, \omega_1}$. Using this implication, we show that the forcing axiom $MM_{\aleph_2}(\aleph_2\mbox{-c.c.})$ is inconsistent. We also derive weak Chang's Conjecture from $MA^{1.5}_{\aleph_2}(\mbox{stratified})$ and use this second implication to give another proof of the inconsistency of $MM_{\aleph_2}(\aleph_2\mbox{-c.c.})$.