Alexander S. Kechris is Professor of Mathematics at the California Institute of Technology. He is the recipient of numerous honors, including the J. S. Guggenheim Memorial Foundation Fellowship and the Carol Karp Prize of the Association for Symbolic Logic. He is also a member of the Scientific Research Board of the American Institute of Mathematics. Benedikt Löwe is Universitair Hoofddocent at the Universiteit van Amsterdam, Professor of Mathematics at the Universität Hamburg and Fellow of Churchill College at the University of Cambridge. He is currently the president of the Deutsche Vereinigung für Mathematische Logik und für Grundlagenforschung der Exakten Wissenschaften (DVMLG) and the Secretary General of the Division for Logic, Methodology and Philosophy of Science and Technology (DLMPST). John R. Steel is Professor of Mathematics at the University of California, Berkeley. Prior to that, he was a professor in the mathematics department at the University of California, Los Angeles. He is a recipient of the Carol Karp Prize of the Association for Symbolic Logic and of a Humboldt Prize. Steel is also a former Fellow at the Wissenschaftskolleg zu Berlin and the Sloan Foundation.

• Preface; Original numbering; Part VII. Extensions of AD, models with choice: 1. A brief history of determinacy Paul B. Larson; 2. “AD plus uniformization” is equivalent to “half $AD_\mathbb{R}$” Alexander S. Kechris; 3. The independence of DC from AD Robert M. Solovay; 4. Games of countable length Donald A. Martin; 5. Some consistency results in ZFC using AD W. Hugh Woodin; 6. Subsets of $\aleph_1$ constructible from a real Alexander S. Kechris; 7. AD and the uniqueness of the supercompact measures on $\wp_{\omega_1}(\lambda)$ W. Hugh Woodin; 8. The extender algebra and $\sum_1^2$-absoluteness Ilijas Farah; Part VIII. Other topics: 9. On Vaught's conjecture John R. Steel; 10. Capacities and analytic sets Claude Dellacherie; 11. More saturated ideals Matthew Foreman; 12. The fourteen Victoria Delfino problems and their status in the year 2020 Andrés Eduardo Caicedo and Benedikt Löwe; Bibliography.

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