ABOUT THIS BOOK
Myaiminthismonographistogiveabriefbutcoherentexpositionofthemainresults
and methods of descriptive set theory. I have made no attempt to be complete; in a
subjectso broad this woulddegenerateinto a long catalogof specializedresults which
wouldcoverupthemainthread. Onthecontrary,Ihavetriedveryhardtobeselective,
so that the central ideas stand out.
Much of the material is in the exercises. A very few of them are simple, to test the
reader’s comprehension, and a few more give interesting extensions of the theory or
sidelines. Thevastmajorityoftheexercisesareanintegralpartofthemonographand
would be normally billed “theorems.” There are extensive “hints” for them, proofs
really, with some of the details omitted.
Ihavetriedhardtoattributealltheimportantresultsandideastothosewhoinvented
them but this was not an easy task and I have undoubtedly made many errors. There
isnosuggestionthatunattributedresultsaremineorarepublishedhereforthefirsttime.
When I do not give credit for something, the most likely explanation is that I could
not determine the correct credit. My own results are immodestly attributed to me,
including those which are first published here.
Many of the references are in the historical sections at the end of each chapter. The
paragraphs of these sections are numbered and the footnotes in the body of the text
refer to these paragraphs—each time meaning the section at the end of the chapter
where the reference occurs. In a first reading, it is best to skip these historical notes
and read them later, after one is familiar with the material in the chapter.
The order of exposition follows roughly the historical development of the subject,
simply because this seemed the best way to do it. It goes without saying that the
classicalresultsarepresentedfromamodernpointofviewandusingmodernnotation.
What appeals to me most about descriptive set theory is that to study it you must
really understandso many things: you need a little bit of topology, analysis and logic,
a good deal of recursive function theory and a great deal of set theory, including
constructibility, forcing, large cardinals and determinacy. What makes the writing of
abookonthesubjectsodifficultisthatyoumustexplainsomanythings: alittlebitof
topology, analysis and logic, a good deal of recursive function theory, etc. Of course,
one could aim the book at those who already know all the prerequisites, but chances
are that these few potential readers already know descriptive set theory. My aim has
been to make this material accessible to a mathematician whose particular field of
specializationcouldbeanything, butwhohasaninterestinsettheory,oratleastwhat
usedtobecalled“thetheoryofpointsets.” Hecertainlyknowswhateverlittletopology
and analysis are required, because he learned that as an undergraduate, and he has
readHalmos’ NaiveSetTheory [1960]orasimilartext. Beyondthat, whatheneedsto
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