Abstract
The generic Vopěnka principle, we prove, is relatively consistent with the ordinals being non-Mahlo. Similarly, the generic Vopěnka scheme is relatively consistent with the ordinals being definably non-Mahlo. Indeed, the generic Vopěnka scheme is relatively consistent with the existence of a \(\Delta _2\)-definable class containing no regular cardinals. In such a model, there can be no \(\Sigma _2\)-reflecting cardinals and hence also no remarkable cardinals. This latter fact answers negatively a question of Bagaria, Gitman and Schindler.
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References
Bagaria, J.: \(C^{(n)}\)-cardinals. Arch. Math. Logic 51(3–4), 213–240 (2012)
Bagaria, J., Brooke-Taylor, A.: On colimits and elementary embeddings. J. Symb. Logic 78(2), 562–578 (2013)
Bagaria, J., Gitman, V., Schindler, R.: Generic Vopěnka’s Principle, remarkable cardinals, and the weak Proper Forcing Axiom. Arch. Math. Logic 56(1–2), 1–20 (2017)
Bagaria, J., Hamkins, J.D., Tsaprounis, K., Usuba, T.: Superstrong and other large cardinals are never Laver indestructible. Arch. Math. Logic 55(1–2), 19–35 (2016). (Special volume in memory of R. Laver)
Foreman, M.: Ideals and generic elementary embeddings. In: Foreman, M., Kanamori, A. (eds.) Handbook of Set Theory, vols. 1, 2, 3, pp. 885–1147. Springer, Dordrecht (2010)
Gitman, V., Hamkins, J.D., Karagila, A.: Kelley–Morse set theory does not prove the class Fodor theorem (in preparation)
Gitman, V., Schindler, R.: Virtual large cardinals. In: The Proceedings of the Logic Colloquium (2015) (to appear)
Hamkins, J.D.: The Vopěnka principle is inequivalent to but conservative over the Vopěnka scheme (manuscript under review)
Schindler, R.-D.: Proper forcing and remarkable cardinals. II. J. Symb. Logic 66(3), 1481–1492 (2001)
Schindler, R.: Remarkable cardinals. In: Geschke, S., Loewe, B., Schlicht, P. (eds.) Infinity, Computability, and Metamathematics, volume 23 of Tributes, pp. 299–308. College Publications, London (2014)
Solovay, R.M., Reinhardt, W.N., Kanamori, A.: Strong axioms of infinity and elementary embeddings. Ann. Math. Logic 13(1), 73–116 (1978)
Tsaprounis, K.: On \(C^{(n)}\)-extendible cardinals (Submitted)
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The research of the second author has been supported by Grant #69573-00 47 from the CUNY Research Foundation. Commentary concerning this paper can be made at http://jdh.hamkins.org/generic-vopenka-ord-not-mahlo
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Gitman, V., Hamkins, J.D. A model of the generic Vopěnka principle in which the ordinals are not Mahlo. Arch. Math. Logic 58, 245–265 (2019). https://doi.org/10.1007/s00153-018-0632-5
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DOI: https://doi.org/10.1007/s00153-018-0632-5