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Softcover ISBN:  9781470472108 
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Softcover ISBN:  9781470472108 
Product Code:  ULECT/78 
List Price:  $69.00 
MAA Member Price:  $62.10 
AMS Member Price:  $55.20 
eBook ISBN:  9781470475659 
Product Code:  ULECT/78.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $52.00 
Softcover ISBN:  9781470472108 
eBook ISBN:  9781470475659 
Product Code:  ULECT/78.B 
List Price:  $134.00 $101.50 
MAA Member Price:  $120.60 $91.35 
AMS Member Price:  $107.20 $81.20 

Book DetailsUniversity Lecture SeriesVolume: 78; 2023; 165 ppMSC: Primary 03
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.
ReadershipGraduate students and researchers interested in logic.

Table of Contents

Preliminaries

Determinacy

The Wadge hierarchy

Coding lemmas

Properties of pointclasses

Strong partition cardinals

Suslin sets and uniformization

$\mathsf {AD}^+$

Ordinal determinacy

InfinityBorel sets

Cone measure ultraproducts

Vopěnka algebras

Suslin sets and strong codes

Scales from uniformization

Real determinacy from scales

Questions


Additional Material

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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.
Graduate students and researchers interested in logic.

Preliminaries

Determinacy

The Wadge hierarchy

Coding lemmas

Properties of pointclasses

Strong partition cardinals

Suslin sets and uniformization

$\mathsf {AD}^+$

Ordinal determinacy

InfinityBorel sets

Cone measure ultraproducts

Vopěnka algebras

Suslin sets and strong codes

Scales from uniformization

Real determinacy from scales

Questions