Square and Delta reflection

TL;DR

This paper constructs a model of set theory where a specific large cardinal successor satisfies both a strong reflection principle and a square principle, demonstrating their compatibility.

Contribution

It shows that the large cardinal successor leph_{\u02e1_{ ext{omega}^2+1}} can simultaneously satisfy elta-reflection and ircle(\u001omega_{ ext{omega}^2+1}), revealing their compatibility.

Findings

leph_{\u02e1_{ ext{omega}^2+1}} satisfies elta-reflection.

leph_{_{ ext{omega}^2+1}} satisfies ircle(_{ ext{omega}^2+1}).

Both principles can hold simultaneously in the same model.

Abstract

Starting from infinitely many supercompact cardinals, we force a model of ZFC where $\aleph_{\omega^2+1}$ satisfies simultaneously a strong principle of reflection, called $\Delta$-reflection, and a version of the square principle, denoted $\square(\aleph_{\omega^2+1}).$ Thus we show that $\aleph_{\omega^2+1}$ can satisfy simultaneously a strong reflection principle and an anti-reflection principle.