# Papers and notes

All my notes can also be found on Github, which includes the tex files.

Paper: Games and Ramsey-like cardinals –  blog post • arXiv

We generalise the $\alpha$-Ramsey cardinals introduced in Holy and Schlicht (2018) for cardinals $\alpha$ to arbitrary ordinals, and answer several questions posed in that paper. In particular, we show that $\alpha$-Ramseys are downwards absolute to the core model $K$ for all $\alpha$ of uncountable cofinality, that $\omega$-Ramseys are also strategic $\omega$-Ramsey, and that strategic $\omega_1$-Ramsey cardinals are equiconsistent with measurable cardinals, both by showing that they are measurable in $K$ and that they carry precipitous ideals. We also show that the $n$-Ramseys satisfy indescribability properties and use them to characterise ineffable-type cardinals, as well as establishing connections between the $\alpha$-Ramsey cardinals and the Ramsey-like cardinals introduced in Gitman (2011), Feng (1990) and Sharpe and Welch (2011).

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”.

This is an introduction to constructibility theory. We start off by constructing Gödel’s constructible universe L rigorously, in which we also correct Devlin’s errors in his book in this regard, by implementing a solution proven by Mathias; the details of this is put in the appendix. After having shown basic properties of L, we show that both Global Choice and the Generalized Continuum Hypothesis hold in L, implying that Con(ZF) ⇒ Con(ZFC + GCH). We then dedicate a chapter to giving a glimpse of the rich combinatorial structure of L, proving the negation of Suslins Hypothesis in L via. the use of the combinatorial principle ♦. In the last chapter we prove Scott’s Theorem, stating that there exist no measurable cardinals in L.

In this project we introduce the notions of perfect information games in a set-theoretic context, from where we’ll analyse both the consequences of the determinacy of games as well as showing large classes of games are determined. More precisely, we’ll show that determinacy of games over the reals implies that every subset of the reals is Lebesgue measurable and has both the Baire and perfect set property (thereby contradicting the axiom of choice). Next, Martin’s result on Borel determinacy will be presented, as well as his proof of analytic determinacy from the existence of a Ramsey cardinal. Lastly, we’ll present a certain kind of stochastic games (that is, games involving chance) called Blackwell games, and present Martin’s proof that determinacy of perfect information games imply the determinacy of Blackwell games.

We prove a core model dichotomy by adapting the folklore proof to the setting without the measurable cardinal from [Jensen and Steel, 2013] and [Fernandes, 2018].

Note: Characterisation of scaled pointclasses in $L(\mathbb R)$download

We present the characterisation of the scaled pointclasses in $L(\mathbb R)$ under determinacy hypotheses, given in Steel’s “Scales in L(R)” paper. The proof that $\Sigma^{J_\alpha(\mathbb R)}_1$ has the scale property under $\text{Det}(J_\alpha(\mathbb R))$ is given in full to give an idea of how the scales are constructed, and the rest of the results are then given with references. The characterisation is summed up in the concluding corollary as well as the diagram given in the appendix.

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 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.