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9781470472108 Academic Inspection Copy

Extensions of the Axiom of Determinacy

  • ISBN-13: 9781470472108
  • Publisher: AMERICAN MATHEMATICAL SOCIETY
    Imprint: AMERICAN MATHEMATICAL SOCIETY
  • By Paul B. Larson
  • Price: AUD $178.00
  • Stock: 0 in stock
  • Availability: This book is temporarily out of stock, order will be despatched as soon as fresh stock is received.
  • Local release date: 30/03/2024
  • Format: Undefined (254.00mm X 178.00mm) 165 pages Weight: 142g
  • Categories: Mathematical logic [PBCD]
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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
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