Large cardinals and the continuum

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This project is concerned with the study of large cardinal axioms, forcing with large cardinals, connections between large cardinal axioms and the continuum through forcing axioms and through inner model theory, and applications of set theory to reverse mathematics and monadic decidability. Specific research topics to be addressed include: forcing axioms consistent with large values of the continuum; questions in infinitary combinatorics that are related to large cardinals, particularly on the relationship between the tree property and the singular cardinals hypothesis at small cardinals; iterability for inner models, and indicators of strength for the proper forcing axiom in connection with long extender models; uses of strong induction hypotheses in reverse mathematics; and monadic theories that are affected by set theoretic axioms, including monadic theories of ordinals and the monadic theory of the reals restricted to definable sets.Many of the foundational questions of mathematics can be addressed using strong axioms of set theory. Perhaps the most celebrated instances involve questions about definable subsets of the continuum of real numbers. Even fairly simple properties of these sets, for example whether they admit a robust notion of length, are now known to be dependent on strong axioms of set theory. But much still remains unknown about the connection between strong axioms of set theory and the continuum. The motivating goal for the project is to deepen our understanding of this connection. This requires a deeper understanding of the axioms themselves, and of intermediary principles between these axioms and properties of the continuum: principles that can be obtained (provably or consistently) granted the axioms, and directly affect the continuum. The project seeks to extend work on both fronts, with research into models for the axioms, combinatorial principles on infinite sets, and saturation principles of the universe of sets that affect properties of the continuum.