Extensions of the Axiom of Determinacy
Seiten
2023
American Mathematical Society (Verlag)
978-1-4704-7210-8 (ISBN)
American Mathematical Society (Verlag)
978-1-4704-7210-8 (ISBN)
An expository account of work on strong forms of the Axiom of Determinacy (AD) by a group of set theorists in Southern California, particularly W. Hugh Woodin, aimed at readers with a background in basic set theory. The book's first half reviews necessary background material, the second introduces Woodin's axiom system and presents his analysis.
This is an expository account of work on strong forms of the Axiom of Determinacy (AD) by a group of set theorists in Southern California, in particular by W. Hugh Woodin. The first half of the book reviews necessary background material, including the Moschovakis Coding Lemma, the existence of strong partition cardinals, and the analysis of pointclasses in models of determinacy. The second half of the book introduces Woodin's axiom system $/mathrm{AD}^{+}$ and presents his initial analysis of these axioms. These results include the consistency of $/mathrm{AD}^{+}$ from the consistency of AD, and its local character and initial motivation. Proofs are given of fundamental results by Woodin, Martin, and Becker on the relationships among AD, $/mathrm{AD}^{+}$, the Axiom of Real Determinacy, and the Suslin property. Many of these results are proved in print here for the first time. The book briefly discusses later work and fundamental questions which remain open. The study of models of $/mathrm{AD}^{+}$ is an active area of contemporary research in set theory.
The presentation is aimed at readers with a background in basic set theory, including forcing and ultrapowers. Some familiarity with classical results on regularity properties for sets of reals under AD is also expected.
This is an expository account of work on strong forms of the Axiom of Determinacy (AD) by a group of set theorists in Southern California, in particular by W. Hugh Woodin. The first half of the book reviews necessary background material, including the Moschovakis Coding Lemma, the existence of strong partition cardinals, and the analysis of pointclasses in models of determinacy. The second half of the book introduces Woodin's axiom system $/mathrm{AD}^{+}$ and presents his initial analysis of these axioms. These results include the consistency of $/mathrm{AD}^{+}$ from the consistency of AD, and its local character and initial motivation. Proofs are given of fundamental results by Woodin, Martin, and Becker on the relationships among AD, $/mathrm{AD}^{+}$, the Axiom of Real Determinacy, and the Suslin property. Many of these results are proved in print here for the first time. The book briefly discusses later work and fundamental questions which remain open. The study of models of $/mathrm{AD}^{+}$ is an active area of contemporary research in set theory.
The presentation is aimed at readers with a background in basic set theory, including forcing and ultrapowers. Some familiarity with classical results on regularity properties for sets of reals under AD is also expected.
Paul B. Larson, Miami University, Oxford, OH.
Preliminaries: Determinacy
The Wadge hierarchy
Coding lemmas
Properties of pointclasses
Strong partition cardinals
Suslin sets and uniformization
$/mathsf{AD}^+$: Ordinal determinacy
Infinity-Borel sets
Cone measure ultraproducts
Vopenka algebras
Suslin sets and strong codes
Scales from uniformization
Real determinacy from scales
Questions
Bibliography
Index
Erscheinungsdatum | 15.12.2023 |
---|---|
Reihe/Serie | University Lecture Series ; 78 |
Verlagsort | Providence |
Sprache | englisch |
Maße | 178 x 254 mm |
Gewicht | 142 g |
Themenwelt | Mathematik / Informatik ► Mathematik ► Logik / Mengenlehre |
ISBN-10 | 1-4704-7210-4 / 1470472104 |
ISBN-13 | 978-1-4704-7210-8 / 9781470472108 |
Zustand | Neuware |
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