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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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