Breadcrumb
More on Supercompactness and Failures of GCH
Tue 22 January 2013, 17:00
Brent Cody
Fields Institute, Toronto
Organisers: Philip Welch, Peter Holy
ABSTRACT
Easton proved that over ZFC the continuum function on the regular cardinals is highly maleable by showing that any reasonable function can be forced to equal the continuum function. Adding large cardinal axioms to ZFC puts additional restrictions on the possible behaviors of the continuum function. For example, if $\kappa$ is a supercompact cardinal and
GCH holds below $\kappa$ then GCH must hold everywhere. The motivating question will be, given some large cardinal $\kappa$, how can we control the behavior of the continuum function on the regular cardinals by forcing while preserving large cardinal properties of $\kappa$? By surgically modifying a generic filter, Woodin proved that the existence of a measurable cardinal $\kappa$ at which GCH fails is equiconsistent with the existence of a an elementary embedding $j:V\longrightarrow M$ with critical point $\kappa$ such that $M^\kappa\subseteq M$ and $j(\kappa)>\kappa^{++}$. I will present some generalizations of this result concerning
supercompactness and sketch a proof of the most recent result, which is joint work with Menachem Magidor.