PREFACE TO THE SECOND EDITION
There was no question of “updating” this book nearly thirty years after it was first
published—in 1980, volume 100 in the Studies in Logic series of North Holland. The
only completely rewritten sections are 6F, which gives a proof of the determinacy of
Borel sets (a version of Martin’s second proof not available in 1980) and 7F, where
the question of how much choice is needed (especially) to prove Borel determinacy is
examined. There is also a new, brief section 3I on the relativization method of proof,
which has baffled some of the not-so-logicallyminded readers. Beyond that, the main
improvements over the first edition are that
- this one has many fewer errors (I hope);
- the bibliography has been completed and expanded with a small selection of
relevant, more recent publications;
- and many passages have been rewritten.
(It has been said that the most basic instinct in man is not for food or sex but to edit
someone else’s writing—and the urge to edit one’s own writing is, apparently, even
stronger.)
There have been two major developments in Descriptive Set Theory since 1980
which have fundamentally changed the subject.
One is the establishment of a robust connection between determinacy hypotheses,
large cardinal axioms and inner model theory, starting with Martin and Steel [1988]
and Woodin [1988], to such an extent that one cannot now understand any of these
partsofsettheorywithoutalsounderstandingtheothers. Ihaveaddedsome“forward
references”tothesedevelopmentswhentheytouchonquestionsthatwereformulated
in the book.
The other is the explosion in applications of Descriptive Set Theory to other parts
ofmathematics,cf.Kechris[1995]. ThisareareallytookoffwithHarrington,Kechris,
and Louveau [1990] which (with the work that followed it) established the study of
definable equivalence relations on Polish spaces as a subject of its own, with deep
connections to classical mathematics. It was not possible to point to this work in
this revision, especially as the basic result in Silver [1980] was not (for some reason)
included in the original.
Many ofthe notionsand techniquesintroducedin this bookhavebeenusedheavily
in these developments, notably scales and the application of effective methods to the
“classical”theory. Someofithasbecomeobsolete,ofcourse;butIdonotbelievethat
its self-contained, foundationally motivated and unified introduction to the effective
theory and the consequences of determinacy hypotheses has been duplicated.
I am grateful to all those who have sent me comments and corrections, including
(fromtheincompleterecordsthatIhave)BenMiller,MikeBrady,VassilisGregoriades,
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