Inner-Model Reflection Principles

Neil Barton, Andrés Eduardo Caicedo, Günter Fuchs, Joel David Hamkins, Jonas Reitz, Ralf Schindler

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 $$W\subsetneq V$$. 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.

Locations

• Studia Logica - View - PDF
• arXiv (Cornell University) - View - PDF
• KOPS (University of Konstanz) - View - PDF
• PhilPapers (PhilPapers Foundation) - View - PDF
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