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arXiv · 1706.00843

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

Victoria Gitman·Joel David Hamkins
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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 $Δ_2$-definable class containing no regular cardinals. In such a model, there can be no $Σ_2$-reflecting cardinals and hence also no remarkable cardinals. This latter fact answers negatively a question of Bagaria, Gitman and Schindler.

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Victoria Gitman, Joel David Hamkins. 2018-04-15. A model of the generic Vopěnka principle in which the ordinals are not Mahlo. https://arxiv.org/abs/1706.00843

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