Am I allowed to say "first-order Vopěnka cardinal"?

Question

For a cardinal $\kappa$ such that $V_{\kappa}$ satisfies Vopěnka's principle as a first-order axiom schema, am I allowed to say "first-order Vopěnka cardinal", or is there any kind of standard term for it?

Answer 1

In my paper, The Vopěnka principle is inequivalent to but conservative over the Vopěnka scheme, I distinguish between the (second-order) Vopěnka principle and the first-order version, which I call the Vopěnka scheme, and prove that these principles are not equivalent, although they are equiconsistent and moreover, the Vopěnka principle is conservative over the Vopěnka scheme for first-order assertions.

In the paper, I say that a Vopěnka cardinal is a cardinal $\kappa$ such that $V_\kappa$ satisfies the Vopěnka principle, and a Vopěnka scheme cardinal is a cardinal such that $V_\kappa$ satisfies the Vopěnka scheme. Although as I mentioned the Vopěnka principle and Vopěnka scheme are equiconsistent, nevertheless Corollary 10 in the paper shows in contrast that Vopěnka cardinals are strictly stronger in consistency strength than even a closed unbounded proper class of Vopěnka scheme cardinals.

So you are referring to what I call the Vopěnka scheme cardinals.