Descriptive inner model theory (Q2837761)

One of the major advances in pure set theory in recent years is \textit{G. Sargsyan}'s dissertation, titled [A tale of hybrid mice. Berkeley: University of California (PhD Thesis) (2009)]. This paper surveys the recent developments in descriptive inner model theory (DIMT) up to the time his dissertation was completed, in particular his proofs of instances of the \textit{mouse set conjecture} (MSC).NEWLINENEWLINE\hskip1em Given a sentence in the language of set theory (for instance, PFA), a typical problem in modern set theory is to determine its large cardinal strength. For the direction of obtaining the large cardinal lower bound, one often needs to construct canonical inner models for large cardinals. For that, the descriptive inner model theoretical approach is to extract large cardinals from canonical structures of determinacy models, in particular of models for theories in the Solovay hierarchy (\S2). Behind the DIMT approach, there is the philosophy that the Solovay hierarchy covers all levels of the consistency strength hierarchy. Recent progress indicates that this approach together with Woodin's core model induction are very likely the keys to finding the large cardinal lower bound of PFA.NEWLINENEWLINE\hskip1em At the heart of DIMT is the MSC, which conjectures that the most complicated form of definability (the universally Baire sets, to be precise) can be captured by suitable mice (which are all \textit{hod mice} in this paper). The author gives a succinct account of the theory of mice (\S1) and sketched the use of HOD analysis (\S3) of determinacy models in his proofs of MSC instances. The main result discussed in this paper is the following theorem.NEWLINENEWLINE\smallskip \noindent { Main theorem.} Each of the following four statements implies that there is an inner model containing the reals and satisfying AD\({}_{\mathbb R}+ {}\)``\(\Theta\) is regular''. {\parindent=0.7cm\begin{itemize}\item[(1)] CH\({}+{}\)``for some stationary \(S \subseteq \omega_1\) the non-stationary ideal restricted to \(S\) is \(\omega_1\)-dense and super homogeneous''. \item[(2)] Mouse Capturing fails in some inner model \(M\) containing reals and modeling AD\({}^{+}+{}\)``\(V=L({\mathcal P}({\mathbb R}))\)''. \item[(3)] There are divergent models of AD\({^{+}}\). \item[(4)] There is a Woodin limit of Woodin cardinals. NEWLINENEWLINE\end{itemize}} In contrary to the common belief that the strength of AD\({}_{\mathbb R}+ {}\)``\(\Theta\) is regular'' is close to one supercompact cardinal, item (4) above significantly reduces the strength down to a Woodin limit of Woodin cardinals.