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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