Inner-model reflection principles

TL;DR

This paper introduces the inner-model and ground-model reflection principles, exploring their consistency, implications, and relationships with large cardinals, class forcing, and maximality principles in set theory.

Contribution

It defines new reflection principles in set theory, analyzes their consistency with ZFC, and connects them to large cardinals and forcing axioms.

Findings

Inner-model reflection is equiconsistent with ZFC.

These principles are forceable by class forcing.

Inner-model reflection follows from large cardinal assumptions.

Abstract

We introduce and consider the inner-model reflection principle, which asserts that whenever a statement $\varphi(a)$ in the first-order language of set theory is true in the set-theoretic universe $V$, then it is also true in a proper inner model $W\subsetneq V$. A stronger principle, the ground-model reflection principle, asserts that any such $\varphi(a)$ true in $V$ is also true in some non-trivial ground model of the universe with respect to set forcing. These principles each express a form of width reflection in contrast to the usual height reflection of the L\'evy-Montague reflection theorem. They are each equiconsistent with ZFC and indeed $\Pi_2$-conservative over ZFC, being forceable by class forcing while preserving any desired rank-initial segment of the universe. Furthermore, the inner-model reflection principle is a consequence of the existence of sufficient large cardinals, and lightface formulations of the reflection principles follow from the maximality principle MP and from the inner-model hypothesis IMH. We also consider some questions concerning the expressibility of the principles.