Early investigations of the Borel degrees ∗ William W. Wadge Abstract 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. As already indicated, the approach will be technical but fairly informal. I will skip many precise definitions and statements of results; firstly, because the details can take up precious space and obscure the important issues; and secondly, because these detailed formulations can be found elsewhere. Readers who need more precise formulations can find them in the dissertation itself [18], which, if all goes to plan, will soon be published as a book. 2 Definability My research grew out of a seminar Prof. J.W. Addison, Jr. gave in the theory of definability in the Fall of 1967, at UC Berkeley. The theory of definability (founded, according to Addison [2], by Tarski) studies the relationship between the grammatical complexity of definitions and the semantic complexity of the objects (typically sets) that they define. A perfect example is the theorem that formulas whose sets of models are closed under ordinary extension are exactly those equivalent to existential formulas. In definability, it is usually easy to show that an object has a definition of a certain degree of complexity—just come up with it. However, proving the contrary—that no such definition exists— can be extremely difficult. Many of the most important results of definability theory help with this problem by reducing proving nonexistence of a definition to proving existence (of something else). For example, we can prove that there does not exist an existential equivalent of a formula by proving that there exists a model of the formula that can be extended to a model of its negation. * To appear in Proc. of the Cabal Seminar, LNML 37, Springer. 1