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 Wsubsetneq 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 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.
Highlights
Every set theorist is familiar with the classical Levy–Montague reflection principle, which explains how truth in the full set-theoretic universe V reflects down to truth in various rank-initial segments Vθ of the cumulative hierarchy
(1) The inner-model reflection principle asserts that if a statement φ(a) in the first-order language of set theory is true in the set-theoretic universe V, there is a proper inner model W, a transitive class model of ZF containing all ordinals and with a ∈ W V, in which φ(a) is true
In light of the enumeration theorem, the ground-model reflection principle is expressed in the first-order language of set theory as the following schema:
Summary
Every set theorist is familiar with the classical Levy–Montague reflection principle, which explains how truth in the full set-theoretic universe V reflects down to truth in various rank-initial segments Vθ of the cumulative hierarchy. It is stated in ZFC as the existence of such set functions for each set-sized collection of non-empty sets This difference plays no role here, and even the weakening of GBC that uses the ZFC version of choice suffices to express the inner-model reflection principle as indicated above.). The ground-model reflection principle, in contrast, is expressible as a schema in the first-order language of set theory. In light of the enumeration theorem, the ground-model reflection principle is expressed in the first-order language of set theory as the following schema:. In particular we show in Theorems 2.1 and 2.2 that both the lightface and boldface versions of the ground-model reflection principle are obtainable from models of ZFC using forcing constructions. In Theorem 3.8 we show that fine-structural inner models of sufficiently strong large cardinal assumptions satisfy the ground-model reflection principle.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
