On one level, the book under review is, as the author indicates in his very brief concluding chapter, a defense of the iterative conception of set. But exposition of that conception and its direct defense against criticisms take up only two of its seven substantive chapters. For the defense mainly consists of chapter-length examinations of rival conceptions, denominated the naïve, the limitation-of-size, the stratified, and the graph conceptions, examinations that begin in each case with the examiner not unsympathetic, but finish with him unconvinced, so that the iterative conception wins in the end by what he calls ‘inference to the best conception’. It is the book’s role as a one-stop source for both technical information and philosophical commentary on the various conceptions just listed, never before treated at length and with comparisons all in one place, that makes the book an indispensable reference work for anyone interested in the philosophy of set theory. But though the book is much more thorough than, say, [Holmes, 2017], the article on alternative set theories in the Stanford Encyclopedia, still Incurvati’s work does not pretend to be — well, encyclopedic. There are minor topics he leaves out that in their day excited flashes of interest, notably Vopenkism, about which one used to hear a lot, and more importantly coverage of the topics that are included does not pretend to be exhaustive. On the contrary, one of the book’s virtues is that, for all the information it provides, it shows also how open-ended investigation of the different conceptions remains: almost every paragraph raises further issues for future exploration, and it is bound to generate discussion for years to come.
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