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

TL;DR

The paper demonstrates that the Vopěnka principle and the Vopěnka scheme, while not equivalent, share the same first-order consequences and are equiconsistent, with the principle being conservative over the scheme in set theory.

Contribution

It establishes the precise relationship and conservativity between the Vopěnka principle and the Vopěnka scheme within set theory.

Findings

Vopěnka principle is not equivalent to the Vopěnka scheme.

Both axioms are equiconsistent.

GBC plus the Vopěnka principle is conservative over ZFC plus the Vopěnka scheme.

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.