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arXiv · 2012.11202

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

Joan Bagaria·Trevor Wilson
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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 $n <\omega$.

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Joan Bagaria, Trevor Wilson. 2020-12-21. The Weak Vop\v{e}nka Principle for definable classes of structures. https://arxiv.org/abs/2012.11202

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