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

Archive for Mathematical Logic 58 (1-2):245-265 (2019)
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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 \-definable class containing no regular cardinals. In such a model, there can be no \-reflecting cardinals and hence also no remarkable cardinals. This latter fact answers negatively a question of Bagaria, Gitman and Schindler.
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DOI 10.1007/s00153-018-0632-5
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References found in this work BETA

C (N)-Cardinals.Joan Bagaria - 2012 - Archive for Mathematical Logic 51 (3-4):213-240.
Proper Forcing and Remarkable Cardinals II.Ralf-Dieter Schindler - 2001 - Journal of Symbolic Logic 66 (3):1481-1492.
On Colimits and Elementary Embeddings.Joan Bagaria & Andrew Brooke-Taylor - 2013 - Journal of Symbolic Logic 78 (2):562-578.

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