## Logic Seminar

Philip SinkCornell University
Saturation of the nonstationary ideal and $\delta^1_2$, part 1: iterated generic ultrapowers
Wednesday, October 2, 2019 - 4:00pm
Malott 206

Let $\mathbf{\delta}^1_2$ be the supremum of the lengths of all prewellorderings of $\mathbf{R}$ which are $\mathbf{\Delta}^1_2$. As the cardinality of $\mathbf{\delta}^1_2$ is at most as large as the cardinality of $|\mathrm{R}|$ and since any $\mathbf{\Delta}^1_2$-prewellordering of $\mathbf{R}$ is in $L(\mathbf{R})$, the assertion that $\mathbf{\delta}^1_2 = \omega_2$ can be regarded as an effective failure of the Continuum Hypothesis. This multipart talk will give a proof of the following result due to Hugh Woodin: if the nonstationary ideal on $\omega_1$ is saturated and $\mathscr{P}(\omega_1)^\sharp$ exists, then $\mathbf{\delta}^1_2 = \omega_2$. The first part will develop the basic theory of iterated generic ultrapowers of countable transitive models.