# Papers and notes

We give the main ideas in the proof of $\Theta^{L(\mathbb R)}$ being Woodin in $\text{HOD}^{L(\mathbb R)}$ assuming $\textsf{AD}$, due to Woodin. We first give a full proof of Solovay’s theorem of the measurability of $({\bf\delta^1_1})^{L(\mathbb R)}=\omega_1^V$ in $L(\mathbb R)$, then sketch how the same ideas generalises to $({\bf\delta^2_1})^{L(\mathbb R)}$ and lastly show how (essentially) the same techniques are used to show Woodin’s theorem. This is following Koellner and Woodin’s chapter “Large cardinals from determinacy” in the handbook.

We prove the Hahn-Banach Theorem using the ultrafilter lemma.

We provide a proof of $\textsf{AD}^{L(\mathbb R)}$ from a limit of Woodins with a measurable above, based on the approach in Larson’s book “The Stationary Tower”.

Note: $\bf\Sigma^2_1$-absoluteness – download

This note is a write-up of the theorem independently shown by Woodin and Steel that if there exists a measurable Woodin $\delta$ then $\bf\Sigma^2_1$-sentences are forcing absolute for $\delta$-small forcing notions in which $\textsf{CH}$ holds. If we improve the assumption of a measurable Woodin to the existence of a model with a countable-in-V measurable Woodin and which is iterable in all forcing extensions, the result holds for all set-sized forcing notions. The former result is proven using Woodin’s stationary tower forcing and the latter using Woodin’s genericity iterations.

We modify Welch’ proof of the weak covering lemma at Jónsson cardinals below a Woodin cardinal to the measurable-free context from Jensen and Steel’s “K without the measurable”.