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Inner-Model Reflection Principles

  • Neil Barton
  • Andrés Eduardo CaicedoEmail author
  • Gunter Fuchs
  • Joel David HamkinsEmail author
  • Jonas Reitz
  • Ralf Schindler
Open Access
Article

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.

Keywords

Inner-model reflection principle Ground-model reflection principle 

Mathematics Subject Classification

Primary 03E45 Secondary 03E35 03E55 03E65 

Notes

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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Copyright information

© The Author(s) 2019

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Neil Barton
    • 1
  • Andrés Eduardo Caicedo
    • 2
    Email author
  • Gunter Fuchs
    • 3
    • 4
  • Joel David Hamkins
    • 4
    • 5
    • 6
    Email author
  • Jonas Reitz
    • 7
  • Ralf Schindler
    • 8
  1. 1.Kurt Gödel Research Center for Mathematical LogicViennaAustria
  2. 2.Mathematical ReviewsAnn ArborUSA
  3. 3.MathematicsThe Graduate Center of The City University of New YorkNew YorkUSA
  4. 4.MathematicsCollege of Staten Island of CUNYStaten IslandUSA
  5. 5.Mathematics, Philosophy, Computer ScienceThe Graduate Center of The City University of New YorkNew YorkUSA
  6. 6.University of Oxford, and Sir Peter Strawson Fellow in Philosophy University CollegeOxfordUK
  7. 7.MathematicsNew York City College of Technology of The City University of New YorkBrooklynUSA
  8. 8.Institut für mathematische Logik und Grundlagenforschung Fachbereich Mathematik und InformatikUniversität MünsterMünsterGermany

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