# Papers and notes

Note: From Determinacy to Woodins – download

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.

Note: From Woodins to Determinacy – download

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.

Note: Jónsson covering below a Woodin – download

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”.

MSc thesis: Inner model theory – download

We introduce the field of inner model theory, starting from extenders and potential premice. We prove the basic properties of mice such as comparison, the Dodd-Jensen lemma, condensation and solidity. We construct mice via the Jensen-Steel method of using robust K c -constructions and then use these mice to construct the core model K under the assumption that there is no proper class model with a Woodin cardinal – this is proven without the assumption that there exists a measurable cardinal in V . This construction of K is due to Steel and Jensen in their 2013 article “K without the measurable”.