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Cabal Seminar volumes of the Springer Lecture Notes in Mathematics series [Cabal i, Cabal ii, Cabal iii, Cabal iv], unpublished material, and new papers.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. In this volume, we continue with Parts III and IV of the project: Wadge degrees and pointclasses and Projective ordinals. As in our first volume, each of the parts contains an introductory survey (written by Alessandro Andretta and Alain Louveau for Part III and by Steve Jackson for Part IV) putting the papers into a present-day context.

In addition to the reprinted papers, this volume contains papers by Steel (More measures from AD) and Martin (Projective sets and cardinal numbers) that date back to the period of the original Cabal publications but were not included in the old volumes. Jackson contributed a new paper Regular cardinals without the weak partition property with recent results that fit well with the topic of Part IV. The paper Early investigations of the degrees of Borel sets by Wadge is a historical overview of the process of the development of the basic theory of the Wadge degrees. Table 1 gives an overview of the papers in this volume with their original references.

As emphasized in our first volume, 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.

In this paper, I give an overview/summary of the techniques used, and the results derived, in my 1984 PhD dissertation, Reducibility and Determinateness on the Baire Space. In particular, I focus on the calculation of the order type (and structure) of the collection of degrees of Borel sets.

§1 Introduction. I would like in this article to present a overview of the main results of my PhD dissertation, and of the game and other techniques used to derive them.

My first thought was to print the entire dissertation but I quickly realized that it was too long—about ten times too long! Hopefully, this condensed version will still be useful. In producing such a drastically shortened account, I have omitted detailed proofs, and many less important or intermediate results. Also, the remaining definitions and results are for the most part given informally.

In writing this I have in mind, first, colleagues (whether in Mathematics or Computing) who are not familiar with descriptive set theory but nevertheless would like to learn about “Wadge Degrees”. To make the material accessible to these readers I have included some basic information about, say, Borel sets that will be very familiar to Cabal insiders. However, my hope is that even experts in descriptive set theory may learn something, if not about my results, at least about the manner in which they were discovered. In particular, I would like to give some ‘classic’ notions, such as Boolean set operations, the attention they deserve.

This chapter proposes exercises and projects based on CouchDB, a recent database system which relies on many of the concepts presented so far in this book. In brief:

1. CouchDB adopts a semistructured data model, based on the JSON (JavaScript Object Notation) format; JSON offers a lightweight alternative to XML;

2. A database in CouchDB is schema-less: the structure of the JSON documents may vary at will depending on their specific features;

3. In order to cope with the absence of constraint that constitutes the counterpart of this flexibility, CouchDB proposes an original approach, based on structured materialized views that can be produced from document collections;

4. Views are defined with the MapReduce paradigm, allowing both a parallel computation and incremental maintenance of their content;

5. Finally, the system aspects of CouchDB illustrate most of the distributed data management techniques covered in the last part of the present book: distribution based on consistent hashing, support for data replication and reconciliation, horizontal scalability, parallel computing, and so forth.

CouchDB is representative of the emergence of so-called key-value store systems that give up many features of the relational model, including schema, structured querying, and consistency guarantees, in favor of flexible data representation, simplicity and scalability. It illustrates the “No[tOnly]SQL” trend with an original and consistent approach to large-scale management of “documents” viewed as autonomous, rich pieces of information that can be managed independently, in contrast with relational databases, which take the form of a rich graph of interrelated flat tuples.