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arXiv · 1606.03778

The Vop\v{e}nka principle is inequivalent to but conservative over the Vop\v{e}nka scheme

Joel David Hamkins
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Abstract

The Vop\v{e}nka principle, which asserts that every proper class of first-order structures in a common language admits an elementary embedding between two of its members, is not equivalent over GBC to the first-order Vop\v{e}nka scheme, which makes the Vop\v{e}nka assertion only for the first-order definable classes of structures. Nevertheless, the two Vop\v{e}nka axioms are equiconsistent and they have exactly the same first-order consequences in the language of set theory. Specifically, GBC plus the Vop\v{e}nka principle is conservative over ZFC plus the Vop\v{e}nka scheme for first-order assertions in the language of set theory.

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Joel David Hamkins. 2016-06-12. The Vop\v{e}nka principle is inequivalent to but conservative over the Vop\v{e}nka scheme. https://arxiv.org/abs/1606.03778

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