# The equivalence of Woodinized supercompact cardinals and Vopenka cardinals

## Norman Perlmutter

### 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.

Norman Lewis Perlmutter grew up in Toledo, Ohio, earned his bachelor’s degree in mathematics at Grinnell College in Grinnell, Iowa, in 2007, and earned his Ph.D. in mathematics at the CUNY Graduate Center in 2013 under the supervision of Joel David Hamkins. After a year as a visiting assistant professor at Florida Atlantic University, he returned to New York City and to CUNY, taking a position as an assistant professor of mathematics at LaGuardia Community College in 2014. Besides mathematics, his interests include theater, board games, food, travel, and science fiction.