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.

The series is devoted to the publication of high-level monographs on all areas of mathematical logic and its applications. It is addressed to advanced students and research mathematicians, and may also serve as a guide for lectures and for seminars at the graduate level.

Working within the Zermelo-Frankel Axioms of set theory, we will introduce two important contradictory axioms: Axiom of Choice and Axiom of Determinacy. We will explore perfect polish spaces and games on these spaces to see that the Axiom of Determinacy is inconsistent with the Axiom of Choice. We will see some of the major consequences of accepting the Axiom of Determinacy and how some of these results change when accepting the Axiom of Choice. We will consider 2-player games of perfect information wherein we will see some powerful results having to do with properties of the real numbers. We will use a game to illustrate a weak proof of the continuum hypothesis.

This book giving an exposition of the foundations of modern measure theory offers three levels of presentation: a standard university graduate course, an advanced study containing some complements to the basic course, and, finally, more specialized topics partly covered by more than 850 exercises with detailed hints and references. Bibliographical comments and an extensive bibliography with 2000 works covering more than a century are provided.

Comprehensive and self-contained text examines the axiom's relative strengths and consequences, including its consistency and independence, relation to permutation models, and examples and counterexamples of its use. 1973 edition.