Brent Cody: Easton’s Theorem for Woodin cardinals

Friday, July 06 at 1:30pm
Fields institute, Stewart library

Speaker: Brent Cody (Fields)

Title: Easton’s Theorem for Woodin cardinals

Abstract: Easton proved that the continuum function $\kappa\mapsto 2^\kappa$ on regular cardinals can be forced to behave in any way that is consistent with Konig’s Theorem ($\kappa<cf(2^\kappa)$) and monotonicity ($\kappa<\lambda$ implies $2^\kappa\leq 2^\lambda$). In the presence of large cardinals, there are additional restrictions on the possible behaviors of the continuum function on regular cardinals. A natural question to ask is: given a large cardinal $\kappa$, what possible behaviors of the continuum function can we force while preserving the large cardinal property of $\kappa$? I will give a brief outline of the literature in this area. I will also sketch a proof of the following result from my dissertation. Suppose $\delta$ is a Woodin cardinal and $F$ is any class function from the regular cardinals to the cardinals such that

  1. $\delta$ is a closure point of $F$
  2. $\kappa<cf(F(\kappa))$ for each $\kappa\in REG$
  3. $\kappa<\lambda$ implies $F(\kappa)\leq F(\lambda)$ for $\kappa,\lambda\in REG$.

Then there is a cofinality-preserving forcing extension in which $\delta$ remains Woodin and $2^\gamma=F(\gamma)$ for each regular cardinal $\gamma$. The proof uses the tuning fork method of Friedman and Thompson as well as some lifting techniques due to Friedman and Honsik.

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