Robust reflection principles

TL;DR

This paper investigates the robustness of reflection properties of large cardinals, such as stationary reflection and the tree property, under forcing, and introduces strengthened principles that are always robust and hold at large cardinals.

Contribution

It introduces natural strengthenings of reflection principles that are always robust and explores their potential at small cardinals, especially successors of singulars.

Findings

Strengthenings of reflection principles are always robust.

These strengthenings hold at sufficiently large cardinals.

Potential for these principles at small cardinals is examined.

Abstract

A cardinal $\lambda$ satisfies a property P robustly if, whenever $\mathbb{Q}$ is a forcing poset and $|\mathbb{Q}|^+ < \lambda$, $\lambda$ satisfies P in $V^{\mathbb{Q}}$. We study the extent to which certain reflection properties of large cardinals can be satisfied robustly by small cardinals. We focus in particular on stationary reflection and the tree property, both of which can consistently hold but fail to be robust at small cardinals. We introduce natural strengthenings of these principles which are always robust and which hold at sufficiently large cardinals, consider the extent to which these strengthenings are in fact stronger than the original principles, and investigate the possibility of these strengthenings holding at small cardinals, particularly at successors of singular cardinals.