A model of the generic Vop\v{e}nka principle in which the ordinals are not Mahlo

TL;DR

This paper demonstrates the relative consistency of the generic Vopěnka principle with the existence of non-Mahlo ordinals, showing that certain large cardinal assumptions can be avoided in this context.

Contribution

It constructs a model where the generic Vopěnka principle holds without the ordinals being Mahlo, addressing a question about the necessity of Mahlo cardinals.

Findings

The generic Vopěnka principle is consistent with non-Mahlo ordinals.

The generic Vopěnka scheme can be consistent with a definably non-Mahlo class.

No $ ext{Sigma}_2$-reflecting or remarkable cardinals exist in the constructed model.

Abstract

The generic Vop\v{e}nka principle, we prove, is relatively consistent with the ordinals being non-Mahlo. Similarly, the generic Vop\v{e}nka scheme is relatively consistent with the ordinals being definably non-Mahlo. Indeed, the generic Vop\v{e}nka 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.