Inner-Model Reflection Principles

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.

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Acknowledgements

The authors would like to thank Philip Welch for his interest in this project, helpful comments and suggestions. Thanks are also due to the anonymous referee for their valuable feedback. Neil Barton is very grateful for the generous support of the FWF (Austrian Science Fund) through Project P 28420 (The Hyperuniverse Programme). Gunter Fuchs was supported in part by PSC-CUNY Grant 60630-00 48. Ralf Schindler gratefully acknowledges support by the DFG Grant SCHI 484/8-1, “Die Geologie Innerer Modelle”.

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Correspondence to Andrés Eduardo Caicedo or Joel David Hamkins.

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This article grew out of an exchange held by some of the authors on the Mathematics.StackExchange site in response to an inquiry posted by the first-named author concerning the nature of width-reflection in comparison to height-reflection [2]. Commentary concerning this paper can be made at http://jdh.hamkins.org/inner-model-reflection-principles.

Presented by Heinrich Wansing

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Barton, N., Caicedo, A.E., Fuchs, G. et al. Inner-Model Reflection Principles. Stud Logica 108, 573–595 (2020). https://doi.org/10.1007/s11225-019-09860-7

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Keywords

  • Inner-model reflection principle
  • Ground-model reflection principle

Mathematics Subject Classification

  • Primary 03E45
  • Secondary 03E35
  • 03E55
  • 03E65