Inner-Model Reflection Principles

Abstract

We introduce and consider the inner-model reflection principle, which asserts that whenever a statement φ(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⊊ V. A stronger principle, the ground-model reflection principle, asserts that any such φ(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 Lévy–Montague reflection theorem. They are each equiconsistent with ZFC and indeed Π 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.