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.
Categories: math.LO