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Indestructibility of Vopěnka’s Principle

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Abstract

Vopěnka’s Principle is a natural large cardinal axiom that has recently found applications in category theory and algebraic topology. We show that Vopěnka’s Principle and Vopěnka cardinals are relatively consistent with a broad range of other principles known to be independent of standard (ZFC) set theory, such as the Generalised Continuum Hypothesis, and the existence of a definable well-order on the universe of all sets. We achieve this by showing that they are indestructible under a broad class of forcing constructions, specifically, reverse Easton iterations of increasingly directed closed partial orders.

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References

  1. Adámek, J., Rosický, J.: Locally Presentable and Accessible Categories. London Mathematical Society Lecture Note Series, vol. 189. Cambridge University Press, Cambridge (1994)

  2. Apter A.W.: Strong cardinals can be fully Laver indestructible. Math. Logic Quart. 48(4), 499–507 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Apter A.W., Hamkins J.D.: Universal indestructibility. Kobe J. Math. 16, 119–130 (1999)

    MathSciNet  MATH  Google Scholar 

  4. Apter A.W., Sargsyan G.: An equiconsistency for universal indestructibility. J. Symb. Log. 75(1), 314–322 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Asperó D., Friedman S.-D.: Large cardinals and locally defined well-orders of the universe. Ann. Pure Appl. Log. 157(1), 1–15 (2009)

    Article  MATH  Google Scholar 

  6. Bagaria, J.: C (n) cardinals, (in preparation)

  7. Baumgartner, J.E.: Iterated Forcing, Surveys in Set Theory. London Mathematical Society Lecture Note Series, vol. 87, pp. 1–59. Cambridge University Press, Cambridge (1983)

  8. Brooke-Taylor A.D.: Large cardinals and definable well-orders on the universe. J. Symb. Log. 74(2), 641–654 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Brooke-Taylor A.D., Friedman S.D.: Large cardinals and gap-1 morasses. Ann. Pure Appl. Log. 159(1–2), 71–99 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Casacuberta C., Scevenels D., Smith J.H.: Implications of large-cardinal principles in homotopical localization. Adv. Math. 197(1), 120–139 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cummings J.: Iterated forcing and elementary embeddings. In: Foreman, M., Kanamori, A. (eds) The Handbook of Set Theory, vol. 2, pp. 775–884. Springer, Berlin (2010)

    Chapter  Google Scholar 

  12. Cummings J., Foreman M., Magidor M.: Squares, scales and stationary reflection. J. Math. Log. 1(1), 35–98 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cummings J., Schimmerling E.: Indexed squares. Israel J. Math. 131(1), 61–99 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Friedman S.D.: Fine Structure and Class Forcing, de Gruyter Series in Logic and Its Applications, vol. 3. de Gruyter, Berlin (2000)

    Google Scholar 

  15. Friedman, S.D.: Large cardinals and L-like universes. In: Andretta, A. (ed.) Set Theory: Recent Trends and Applications, Quaderni di Matematica, vol. 17, pp. 93–110. Seconda Università di Napoli (2005)

  16. Gitik M., Shelah S.: On certain indestructibility of strong cardinals and a question of Hajnal. Arch. Math. Log. 28(1), 35–42 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hamkins J.D.: The lottery preparation. Ann. Pure Appl. Log. 101(2–3), 103–146 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hamkins J.D., Johnstone T.A.: Indestructible strong unfoldability. Notre Dame J. Form. Log. 51(3), 291–321 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jech T.: Set Theory. The Third Millennium Edition, Revised and Expanded. Springer, Berlin (2003)

    MATH  Google Scholar 

  20. Jensen, R.B.: Measurable cardinals and the GCH. In: Jech, T.J. (ed.) Axiomatic Set Theory, Proceedings of Symposia in Pure Mathematics, vol. 13 part II, pp. 175–178. American Mathematical Society (1974)

  21. Johnstone T.: Strongly unfoldable cardinals made indestructible. J. Symb. Log. 73(4), 1215–1248 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kanamori, A.: On Vopěnka’s and related principles. In: Macintyre, A., Pacholski, L., Paris, J. (eds.) Logic Colloquium ’77, Studies in Logic and the Foundations of Mathematics, vol. 96, pp. 145–153. North-Holland (1978)

  23. Kanamori A.: The Higher Infinite. Springer, Berlin (2003)

    MATH  Google Scholar 

  24. Kunen K.: Set theory. North-Holland, Amsterdam (1980)

    MATH  Google Scholar 

  25. Laver R.: Making the supercompactness of κ indestructible under κ-directed closed forcing. Israel J. Math. 29(4), 385–388 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lévy A., Solovay R. M.: Measurable cardinals and the continuum hypothesis. Israel J. Math. 5(4), 234–248 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lévy, A.: A Hierarchy of Formulas in Set Theory. Memoirs of the American Mathematical Society, vol. 57 (1965)

  28. Powell, W.C.: Almost huge cardinals and Vopenka’s principle. Notices Am. Math. Soc., 19(5), A–616, abstract (1972)

  29. Pultr A., Trnková V.: Combinatorial, Algebraic and Topological Representations of Groups, Semigroups and Categories. North-Holland Mathematical Library, vol. 22. North-Holland (1980)

    MATH  Google Scholar 

  30. Sato K.: Double helix in large large cardinals and iteration of elementary embeddings. Ann. Pure Appl. Log. 146(2–3), 199–236 (2007)

    Article  MATH  Google Scholar 

  31. Scott D.S.: Measurable cardinals and constructible sets. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. IX(7), 521–524 (1961)

    Google Scholar 

  32. Solovay R.M., Reinhardt W.N., Kanamori A.: Strong axioms of infinity and elementary embeddings. Ann. Math. Log. 13(1), 73–116 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  33. Velleman D.J.: Morasses, diamond, and forcing. Ann. Math. Log. 23(2–3), 199–281 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  34. Velleman D.J.: Simplified morasses. J. Symb. Log. 49(1), 257–271 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  35. Vopěnka P., Pultr A., Hedrlín Z.: A rigid relation exists on any set. Commentationes Mathematicae Universitatis Carolinae 6(2), 149–155 (1965)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, UK

    Andrew D. Brooke-Taylor

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  1. Andrew D. Brooke-Taylor
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Correspondence to Andrew D. Brooke-Taylor.

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This research was conducted at the University of Bristol with support from the Heilbronn Institute for Mathematical Research.

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Brooke-Taylor, A.D. Indestructibility of Vopěnka’s Principle. Arch. Math. Logic 50, 515–529 (2011). https://doi.org/10.1007/s00153-011-0228-9

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