Forcing with Large Cardinals

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Our study of mathematics is carried out by means of rigorous deduction from axioms, i.e., proofs. It is well-known that the results of this study are incomplete, and may lead to statements which cannot be decided from the fundamental principles of mathematics (the ZFC axioms). Set theory is a field of mathematical logic which provides tools to analyze the consistency of mathematical statements. The systematic approach by which consistency results are obtained comes through the construction of alternative mathematical universes in which the consistency of various statements can be examined and validated. This project deals with several such construction methods. It aims to develop new methods to address several important problems in infinitary combinatorics, and to study existing methods and their limitations. The main objects of study are (i) algebras in set theory; (ii) diamond sequences, (iii) the Mitchell order; and (iv) inner models of hereditarily ordinal definable sets (HOD). In connection with (i), the project aims to study different notions of singular stationary, which were introduced by Foreman and Magidor, and make advancements in answering questions regarding Jonsson algebras. In connection with (ii), the project addresses diamond type principles and their interaction with compactness principles and cardinal arithmetic assumptions. In connection with (iii), the project aims to study the connection between forcing theory and inner model theory by studying problems concerning the consistency strength of various Mitchell order structures. In connection with (iv), the project aims to study the extent to which HOD is close to the set-theoretic universe V, and address the HOD-conjecture.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.