# Inner-Model Reflection Principles

Abstract. We introduce and consider the inner-model reflection principle, which asserts that whenever a statement 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 . A stronger principle, the ground-model reflection principle, asserts that any such  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 and indeed -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 and from the inner-model hypothesis . We also consider some questions concerning the expressibility of the principles.

Inner-model reflection principles, with Neil Barton, Andres Caicedo, Gunter Fuchs, Joel David Hamkins and Ralf Schindler, Stud Logica (2019). https://doi.org/10.1007/s11225-019-09860-7.  pdf, arXiv