# Equivalence of Vopenka cardinals and Woodinized supercompact cardinals

Set Theory Seminar
I present a tentative result that Woodinized supercompact cardinals (also known as Woodin for supercompactness cardinals) are equivalent to Vopenka cardinals. This result is vaguely hinted at, though not proven, in Kanamori’s text, and I believe I have worked out the details. A cardinal $\kappa$ is Vopenka iff for every collection of kappa many model-theoretic structures with domains elements of $V_\kappa$, there exists an elementary embedding between two of them. A cardinal $\kappa$ is Woodinized supercompact if it meets the definition of a Woodin cardinal, with strongness replaced by supercompactness. That is to say, for every function $f:\kappa \to \kappa$, there exists a closure point $\delta$ of $f$ and an elementary embedding $j:V \to M$ such that $j(\delta)<\kappa$ and $M$ is closed in $V$ under $j(f)(\delta)$ sequences.