The Weak Vop\v{e}nka Principle for definable classes of structures

TL;DR

This paper analyzes the Weak Vopěnka Principle for classes of structures with varying definability complexity, establishing its equivalence to the existence of certain large cardinals at each level.

Contribution

It provides a detailed level-by-level analysis of WVP, linking each definability level to corresponding large cardinal assumptions, especially $orall n<\omega$ $ ext{Sigma}_n$-strong cardinals.

Findings

WVP for $ ext{Sigma}_2$-definable classes is equivalent to a strong cardinal.

WVP for $ ext{Sigma}_n$-definable classes corresponds to $ ext{Sigma}_n$-strong cardinals.

WVP is equivalent to the existence of $ ext{Sigma}_n$-strong cardinals for all $n<\omega$.

Abstract

We give a level-by-level analysis of the Weak Vop\v{e}nka Principle for definable classes of relational structures (WVP), in accordance with the complexity of their definition, and we determine the large-cardinal strength of each level. Thus, in particular we show that WVP for $\Sigma_2$-definable classes is equivalent to the existence of a strong cardinal. The main theorem shows, more generally, that WVP for $\Sigma_n$-definable classes is equivalent to the existence of a $\Sigma_n$-strong cardinal. Hence, WVP is equivalent to the existence of a $\Sigma_n$-strong cardinal, all $n <\omega$.