Brent Cody earned his Ph.D. under my supervision at the CUNY Graduate Center in June, 2012.  Brent’s dissertation work began with the question of finding the exact consistency strength of the GCH failing at a cardinal $\theta$, when $\kappa$ is $\theta$-supercompact.  The answer turned out to be a $\theta$-supercompact cardinal that was also $\theta^{++}$-tall.  After this, he quickly dispatched more general instances of what he termed the Levinski property for a variety of other large cardinals, advancing his work towards a general investigation of the Easton theorem phenomenon in the large cardinal context, which he is now undertaking.  Brent held a post-doctoral position at the Fields Institute in Toronto, afterwards taking up a position at the University of Prince Edward Island.  He is now at Virginia Commonwealth University.

Brent Cody

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Brent Cody, “Some Results on Large Cardinals and the Continuum Function,” Ph.D. dissertation for The Graduate Center of the City University of New York, June, 2012.

Abstract.  Given a Woodin cardinal $\delta$, I show that if $F$ is any Easton function with $F”\delta\subseteq\delta$ and GCH holds, then there is a cofinality preserving forcing extension in which $2^\gamma= F(\gamma)$ for each regular cardinal $\gamma<\delta$, and in which $\delta$ remains Woodin.

I also present a new example in which forcing a certain behavior of the continuum function on the regular cardinals, while preserving a given large cardinal, requires large cardinal strength beyond that of the original large cardinal under consideration. Specifically, I prove that the existence of a $\lambda$-supercompact cardinal $\kappa$ such that GCH fails at $\lambda$ is equiconsistent with the existence of a cardinal $\kappa$ that is $\lambda$-supercompact and $\lambda^{++}$-tall.

I generalize a theorem on measurable cardinals due to Levinski, which says that given a measurable cardinal, there is a forcing extension preserving the measurability of $\kappa$ in which $\kappa$ is the least regular cardinal at which GCH holds. Indeed, I show that Levinski’s result can be extended to many other large cardinal contexts. This work paves the way for many additional results, analogous to the results stated above for Woodin cardinals and partially supercompact cardinals.