Square principles in Pmax extensions

TL;DR

This paper explores the impact of $ ext{P}_{ ext{max}}$ forcing on square principles at $oldsymbol{\omega_2}$ and $oldsymbol{\omega_3}$, producing models with specific failures of these principles under strong determinacy assumptions.

Contribution

It demonstrates how $ ext{P}_{ ext{max}}$ forcing can create models where square principles at $oldsymbol{\omega_2}$ and $oldsymbol{\omega_3}$ fail, linking determinacy and combinatorial set theory.

Findings

Models where $2^{oldsymbol{\aleph_0}}=2^{oldsymbol{\aleph_1}}=oldsymbol{\aleph_2}$

Failure of $ ext{square}(oldsymbol{\omega_2})$ and $ ext{square}(oldsymbol{\omega_3})$

Forcing with $ ext{P}_{ ext{max}}$ affects combinatorial principles at uncountable cardinals

Abstract

By forcing with $\mathbb{P}_{\rm max}$ over strong models of determinacy, we obtain models where different square principles at $\omega_2$ and $\omega_3$ fail. In particular, we obtain a model of $2^{\aleph_0}=2^{\aleph_1}=\aleph_2 + \lnot\square(\omega_2) + \lnot\square(\omega_3)$.