Developing the theory up to the current state-of-the art, this book studies the minimal model of the Largest Suslin Axiom (LSA), which is one of the most important determinacy axioms and features prominently in Hugh Woodin's foundational framework known as the Ultimate L. The authors establish the consistency of LSA relative to large cardinals and develop methods for building models of LSA from other foundational frameworks such as Forcing Axioms. The book significantly advances the Core Model Induction method, which is the most successful method for building canonical inner models from various hypotheses. Also featured is a proof of the Mouse Set Conjecture in the minimal model of the LSA. It will be indispensable for graduate students as well as researchers in mathematics and philosophy of mathematics who are interested in set theory and in particular, in descriptive inner model theory.
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A thorough study of the minimal model of the Largest Suslin Axiom, including the first proof of its consistency.
1. Introduction;
2. Hybrid J-structures;
3. Short tree strategy mice;
4.
A comparison theory of HOD mice;
5. HOD mice revisited;
6. The internal
theory of LSA HOD mice;
7. Analysis of HOD;
8. Models of LSA as derived
models;
9. Condensing sets;
10. Applications;
11. A proof of square in
LSA-small HOD mice;
12. LSA from PFA; References; Index.
Grigor Sargsyan is Professor at the Institute of Mathematics of the Polish Academy of Sciences. He holds a Ph.D. in mathematics from the University of California, Berkeley. His thesis, which is the precursor of the work presented in this book, received Association of Symbolic Logic's Sacks Prize. He later received the National Science Foundation's Career Award, which was awarded to complete the work carried out in this book. Nam Trang is Assistant Professor of Mathematics at the University of North Texas. He holds a Ph.D. in mathematics from the University of California, Berkeley. He is the recipient of the National Science Foundation's two regular grants and a Career Award, part of which supports the writing of this book. He also received an Outstanding Research Award from the UNT Department of Mathematics.
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