Cardinals Beyond Choice and the HOD-Dichotomy

Treballs Finals del Màster de Lògica Pura i Aplicada, Facultat de Filosofia, Universitat de Barcelona. Curs: 2019-2020. Tutor: Joan BagariaIn the 2019 paper "Large Cardinals Beyond Choice" [1], Bagaria, Koellner and Woodin apply the large cardinal techniques and results developed fromWoodin's work on the HOD-Dichotomy to determine the structural resemblance of HOD to V . Whereas standard inner model theory attempts to nd suitable inner models for large cardinals, this new program is aimed at exploring very large cardinals that \break" the resemblance of HOD to V . This paper attempts to explain in full detail the tools and arguments required for that body of work

Inner models with large cardinal features usually obtained by forcing

We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal \kappa for which 2^\kappa=\kappa^+, another for which 2^\kappa=\kappa^++ and another in which the least strongly compact cardinal is supercompact. If there is a strongly compact cardinal, then there is an inner model with a strongly compact cardinal, for which the measurable cardinals are bounded below it and another inner model W with a strongly compact cardinal \kappa, such that H_{\kappa^+}^V\subseteq HOD^W. Similar facts hold for supercompact, measurable and strongly Ramsey cardinals. If a cardinal is supercompact up to a weakly iterable cardinal, then there is an inner model of the Proper Forcing Axiom and another inner model with a supercompact cardinal in which GCH+V=HOD holds. Under the same hypothesis, there is an inner model with level by level equivalence between strong compactness and supercompactness, and indeed, another in which there is level by level inequivalence between strong compactness and supercompactness. If a cardinal is strongly compact up to a weakly iterable cardinal, then there is an inner model in which the least measurable cardinal is strongly compact. If there is a weakly iterable limit \delta of <\delta-supercompact cardinals, then there is an inner model with a proper class of Laver-indestructible supercompact cardinals. We describe three general proof methods, which can be used to prove many similar results