Sharon Berry, A Logical Foundation for Potentialist Set Theory, Cambridge University Press, 2022, 240pp., $99.99 (hbk), ISBN 9781108834315.
Reviewed by John P. Burgess, Princeton University
The central project of the book under review, occupying the first half of the text, has three components: (I) A critique of mainstream approaches to set theory (with which the book presupposes some familiarity, as will this review), approaches that assume the actual existence of a full cumulative hierarchy of sets, satisfying the axioms of ZFC. (II) The introduction of a new modality, conditional logical possibility. (III) Its use to produce a paraphrase of ZFC (inspired by but improving on the earlier work of Hillary Putnam and Geoffrey Hellman) requiring only the potential existence of (structures isomorphic to) bigger and bigger parts of the cumulative hierarchy, namely, the initial segments called V(α) in the literature, for larger and larger ordinals α. The technicalities come…
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