Inner-model reflection principles

Abstract

We introduce and consider the inner-model reflection principle, which asserts that whenever a statement $\phi (a)$ 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 $\phi (a)$ 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 ${\Pi }_2$-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.

17 pages, revised version incorporating suggestions of the referees; a new co-author has been added. Commentary concerning this paper can be made at http://jdh.hamkins.org/inner-model-reflection-principles

Cited by in corpus (1)

• Set theory with a proper class of indiscernibles