Inter-model reflection principles - Barton, N; Caicedo, AE; Fuchs, G; Hamkins, JD; Reitz, J; Schindler, R (Springer, 2019)
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 Π2-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.