It is only before or after a book is written that it makes sense to talk about the reason for writing it. In between, reasons are as numerous as the days. Looking back, though, I can see some motives that remained more or less constant in the writing of this book and that may not be completely obvious.

I wanted to write a book that would fill what I see as an artificial gap between model theory and recursion theory.

I wanted to write a companion volume to books by two friends, H. J. Keisler's Model Theory for Infinitary Logic and Y.N. Moschovakis’ Elementary Induction on Abstract Structures, without assuming material from either.

I wanted to set forth the basic facts about admissible sets and admissible ordinals in a way that would, at long last, make them available to the logic student and specialist alike. I am convinced that the tools provided by admissible sets have an important role to play in the future of mathematical logic in general and definability theory in particular. This book contains much of what I wish every logician knew about admissible sets. It also contains some material that every logician ought to know about admissible sets.

Several courses have grown out of my desire to write this book. I thank the students of these courses for their interest, suggestions and corrections. A rough first draft was written at Stanford during the unforgettable winter and spring of 1973. The book was completed at Heatherton, Freeland, Oxfordshire during the academic year 1973—74 while I held a research grant from the University of Wisconsin and an SRC Fellowship at Oxford. I wish to thank colleagues at these three institutions who helped to make it possible for me to write this book, particularly Professors Feferman, Gandy, Keisler and Scott. I also appreciate the continued interest expressed in these topics over the past years by Professor G. Kreisel, and the support of the Q-Group during the preparation of this book. I would like to thank Martha Kirtley and Judy Brickner for typing and John Schlipf, Matt Kaufmann and Azriel Levy for valuable comments on an earlier version of the manuscript. I owe a lot to Dana Scott for hours spent helping prepare the final manuscript.