Archive for Mathematical Logic

, Volume 58, Issue 1–2, pp 245–265 | Cite as

A model of the generic Vopěnka principle in which the ordinals are not Mahlo

  • Victoria GitmanEmail author
  • Joel David Hamkins


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.


Virtual large cardinals Remarkable cardinals Generic Vopenka principle 

Mathematics Subject Classification

03E35 03E55 


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematics Program, CUNY Graduate CenterThe City University of New YorkNew YorkUSA
  2. 2.Mathematics, Philosophy, Computer ScienceThe Graduate Center of The City University of New YorkNew YorkUSA
  3. 3.College of Staten Island of CUNYStaten IslandUSA

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