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?



Am I allowed to say "first-order Vopěnka cardinal"?
In my paper, [The Vopěnka principle is inequivalent to but conservative over the Vopěnka scheme](http://jdh.hamkins.org/vopenka-principle-vopenka-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. 


