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

2017/06/02 by Victoria Gitman, Joel David Hamkins, Gitman, Victoria +1 · 1 citation
Mathematics · #Advanced Topology and Set Theory #Homotopy and Cohomology in Algebraic Topology #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.1706.00843

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 Δ₂-definable class containing no regular cardinals. In such a model, there can be no Σ₂-reflecting cardinals and hence also no remarkable cardinals. This latter fact answers negatively a question of Bagaria, Gitman and Schindler.

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