# Topic Archive: Woodin for supercompact

Set theory seminarFriday, September 7, 201212:00 amGC 6417

# The equivalence of Woodinized supercompact cardinals and Vopenka cardinals

LaGuardia Community College, CUNY

I present a tentative result that 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. Kappa is Vopenka iff for every collection of kappa many model-theoretic structures with domain subset of $V_kappa$ there exists an elementary embedding between two of them. Kappa is Woodin for supercompactness if it meets the definition of a Woodin cardinal, with strongness replaced by supercompactness. That is to say, for every function $f:kappatokappa$, there exists a closure point delta of f and an elementary embedding $j:Vto M$ such that $j(delta)ltkappa$ and $latex M$ is closed in $latex V$ under $j(f)(delta)$ sequences.