The notion of reflection plays a central role in modern Set Theory since the descovering of the well-known Lévy and Montague \textit{Reflection principle}. For any $n\in\omega$, let $C^{(n)}$ denote the class of all ordinals $\kappa$ which correctly interprets the $\Sigma_n$-statements of the universe, with parametes in $V_\kappa$.
In 2016, Bagaria introduced the $C^{(n)}$-hierarchy of large cardinals in order to do an exhaustive study of the region between the first supercompact and Vopenka Principle. To be more precise, he showed that there is a nice correspondence between the amount of (what he called) $C^{(n)}$-extendible cardinals and the degree of structural reflection enjoyed by the universe.
Among this principles we can find Vop\v{e}nka Principle which is equivalent to the existence of a $C^{(n)}$-extendible cardinal, for each $n\in\omega$.
In the same spirit, Bagaria introduced the corresponding $C^{(n)}$-version of other classical large cardinal notion and study its consistency strength.
Several study on this aspect has also been done by K. Tsaprounis in a series of papers where many of the questions posed by Bagaria have been settled. However, some central issues on the relation between supercompact, $C^{(n)}$-supercompact and $C^{(n)}$-extendible cardinals remained open. In this talk we will present some recent progress in this regard.
(Joint work with Yair Hayut and Menachem Magidor)

## Date:

Wed, 23/05/2018 - 11:00 to 13:00

## Location:

Ross 63