# Inner-Model Reflection Principles

Barton, N
Caicedo, A
Fuchs, G
Hamkins, J
Reitz, J
Schindler, R

1 June 2020

STUDIA LOGICA

## Last Updated:

2021-02-14T00:11:06.79+00:00

3

108

## DOI:

10.1007/s11225-019-09860-7

573-595

## abstract:

&#xA9; 2019, The Author(s). We introduce and consider the inner-model reflection principle, which asserts that whenever a statement &#x3C6;(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&#x228A; V. A stronger principle, the ground-model reflection principle, asserts that any such &#x3C6;(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 L&#xE9;vy&#x2013;Montague reflection theorem. They are each equiconsistent with ZFC and indeed &#x3A0; 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.

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Submitted

Journal Article