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 Docent in Logic in the Institute for Logic, Language and Computation at the Universiteit van Amsterdam and Professor of Mathematics at the Universität Hamburg. He is the vice-president of the Deutsche Vereinigung für Mathematische Logik und für Grundlagenforschung der Exakten Wissenschaften (DVMLG) and a Managing Editor of the journal Mathematical Logic Quarterly. 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.

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• Preface Alexander S. Kechris, Benedikt Löwe and John R. Steel; Original numbering; Part V. HOD and its Local Versions: Ordinal definability in models of determinacy – Introduction to Part V John R. Steel; Partially playful universes Howard S. Becker; Ordinal games and playful models Yiannis N. Moschovakis; Measurable cardinals in playful models Howard S. Becker and Yiannis N. Moschovakis; Introduction to Q-theory Alexander S. Kechris, Donald A. Martin and Robert M. Solovay; On the theory of ∏1/3 sets of reals, II Alexander S. Kechris and Donald A. Martin; An inner models proof of the Kechris–Martin theorem Itay Neeman; A theorem of Woodin on mouse sets John R. Steel; HOD as a core model John R. Steel and W. Hugh Woodin; Part VI. Recursion Theory: Recursion theoretic papers – Introduction to Part VI Leo A. Harrington and Theodore A. Slaman; On recursion in E and semi-Spector classes Phokion G. Kolaitis; On Spector classes Alexander S. Kechris; Trees and degrees Piergiorgio Odifreddi; Definable functions on degrees Theodore A. Slaman and John R. Steel; ∏1/2 monotone inductive definitions Donald A. Martin; Martin's conjecture, arithmetic equivalence, and countable Borel equivalence relations Andrew Marks, Theodore A. Slaman and John R. Steel; Bibliography.