An application of strenghtening of the Vopenka's principle

Question

The well-known Vopěnka prinicple states that for a proper class of models of a first-order language $L_{\omega\ \omega}$ there exists a nontrivial embedding from one to another model and it has many applications. Suppose that we strengthen this existence and require that there is $\lambda$ many elementary embeddings. Are there any applications of this modified VP, or at least is it consistent with ZFC?

Answer 1

For any cardinal $\lambda$ your suggested strengthening is just equivalent to the usual Vopěnka's principle (and so is consistent with ZFC relative to very large cardinals when formulated in one of its first order versions).

For the nontrivial direction suppose that $\mathsf{VP}$ holds, let $\langle \mathcal M_\alpha\mid\alpha\in\mathrm{Ord}\rangle$ be a proper class of structures of some first order language and let $\lambda$ be a fixed cardinal. Suppose for a contradiction that there are less than $\lambda$-many $\xi\in\mathrm{Ord}$ such that $\mathcal M_\xi$ elementary embeds into some $\mathcal M_\alpha$ and let $$A=\{\xi\in\mathrm{Ord}\mid \exists\alpha\in\mathrm{Ord}\; \mathcal M_\xi\prec\mathcal M_\alpha\}.$$ Then $\langle \mathcal M_\alpha\mid\alpha\in\mathrm{Ord}\setminus A\rangle$ is a proper class of structures of the same language no two of which elementary embed into each other, contradicting $\mathsf{VP}$.