This book continues the series of volumes containing reprints of the papers in the original Cabal Seminar volumes of the Springer Lecture Notes in Mathematics series [Cabal i, Cabal ii, Cabal iii, Cabal iv], unpublished material, and new papers. The first volume, [Cabal I], contained papers on games, scales and Suslin cardinals. The second volume, [Cabal II], contained papers on Wadge degrees and pointclasses and projective ordinals. In this volume, we continue with Parts V and VI of the project: Ordinal definability in models of determinacy and Recursion theory. As in our first two volumes, each of the parts contains an introductory survey (written by John Steel for Part V and by Leo Harrington and Ted Slaman for Part VI) putting the papers into a present-day context.
In addition to the reprinted papers, this volume contains a paper by Kechris and Martin (On the theory of Π13sets of reals, II) that dates back to the period of the original Cabal publications but was not included in the old volumes. Neeman contributed a new paper, An inner models proof of the Kechris–Martin theorem, related to this paper. Steel and Woodin contributed two new papers (A theorem of Woodin on mouse sets, authored by Steel, and HODas a core model, jointly) with recent results that fit well with the topics of Part V. There is also a new paper by Marks, Slaman and Steel (Martin's conjecture, arithmetic equivalence, and countable Borel equivalence relations) that contains earlier, unpublished, as well as new results related to the theme of Part VI. Table 1 gives an overview of the papers in this volume with their original references.
As emphasized in our first two volumes, our project is not to be understood as a historical edition of old papers. In the retyping process, we uniformized and modernized notation and numbering of sections and theorems. As a consequence, references to papers in the old Cabal volumes will not always agree with references to their reprinted versions. In this volume, references to papers that already appeared in reprinted form will use the new numbering. In order to help the reader to easily cross-reference old and new numberings, we provide a list of changes after the preface.