Conceptions of Set and the Foundations of Mathematics
Conceptions of Set and the Foundations of Mathematics appeared with Cambridge University Press in 2020. Available at: CUP | Amazon
Overview
Sets are central to mathematics and its foundations, but what are they? This book provides a detailed examination of all the major conceptions of set and discusses their virtues and shortcomings, as well as introducing the fundamentals of the alternative set theories with which these conceptions are associated. It shows that the conceptual landscape includes not only the naïve and iterative conceptions but also the limitation of size conception, the definite conception, the stratified conception and the graph conception. In addition, the book presents a novel, minimalist account of the iterative conception which does not require the existence of a relation of metaphysical dependence between a set and its members.
Reviews
'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 ... 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.'
John Burgess, Philosophia Mathematica
'without doubt the best discussion available of the variety of conceptions of set that have been proposed.'
'...explores different approaches to the concept of set, comparing and evaluating these approaches in terms of how they might provide such a foundation. Along the way, it offers an excellent, self-contained introduction to a wide range of important topics in the philosophy of set theory, while at the same time, proposing a novel framework within which one can evaluate different conceptions of set.'