Contributed Talk

# Disentangling Sigma_2 variants of supercompactness

## Andrew Brooke-Taylor

Tuesday, 18:10, J335

If you know what a supercompact cardinal is, there are many possibilities
that come to mind for what the phrase “\(\Sigma_2\)-supercompact” could mean.
Noting that \(\textrm{HOD}\) is \(\Sigma_2\)-definable, it is natural to throw
\(\textrm{HOD}\)-supercompactness and supercompactness in \(\textrm{HOD}\) into
this mix, particularly given the interest in large cardinals in \(\textrm{HOD}\)
arising from Woodin’s \(\textrm{HOD}\) conjecture.

We report on joint work with Shoshana Friedman
disentangling these various possibilities,
in part building on work of Bea Adam-Day [1], with models separating
examples of these variants that are inequivalent.

## Bibliography

- Beatrice Adam-Day,
,PhD Thesis, University of Leeds, 2024.*Indestructibility and \(C^{(n)}\)-Supercompact Cardinals*