Set Theory and Its Philosophy: A Critical Introduction

I was merely trying to find a book that discusses philosophy of mathematics related to set theory, and I found this book. The book /is not/ and /should not/ be regarded as an introduction to axiomatic set theory. It is, rather, a not-that-technical (due to the fact that a formalization of set theory involves nothing that is too much technical), expositional monograph on the newly developed /Scott-Potter set theory/ $\mathsf{ZU}$ and its variants such as $\mathsf{ZfU}$. For students who really want to be introduced into axiomatic set theory in its currently widely used form, a book on $\mathsf{ZFC}$ should be used instead.

The system $\mathsf{ZU}$, centered around the notion of /cumulative iterative hierarchy/, has several advantages over the usual $\mathsf{ZFC}$, in that 1. The cumulative (iterative) hierarchy is no longer parasitic upon the notion of ordinal to be defined, avoiding the circularity that plagues $\mathsf{ZFC}$. Levels are defined first, and an ordinal is a set in a particular level. 2. The Axiom of Choice is no longer needed in order that ordinals are useful. On the basis of $\mathsf{ZF}$, $\mathsf{C}$ is needed in order that for every set $X$ there's an ordinal $\alpha$ and a bijection $\alpha \to X$. For $\mathsf{ZU}$, choice is not needed anymore, due to Scott's trick. The first point is particularly impressive, since a strong argument against cumulative hierarchy, as given by Priest and others concerning the notion of ordinals as sets to be pre-defined in order that what counts as a set is to be defined, /e.g./

+begin_quote

First, it is not even clear that the notion of the cumulative hierarchy makes sense without a prior and different notion of set. For it is clear that the construction is parasitic upon a prior notion of ordinal. Until we have specified ‘‘how long’’ the construction is to go on, that is, how far the ordinals extend, the cumulative hierarchy is ill defined.

+end_quote

is demolished, which further means that Godel's corresponding comment in his 1933 unpublished article /The present situation in the foundation of mathematics/ can actually be valid.

The book, as indicated by /and its philosophy/ in its title, also offers extensive philosophical and historical comments on the rationales behind each step made, either it be the choice of axioms or the strategies adopted in defining sets, /i.e./ "Potter does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true." But it's not introductory and requires a background in philosophy of mathematics more than that can be offered by, for example, Shapiro's introductory book. Nevertheless the discussions that Potter puts forward, notes given in the end of each sections, and the literature cited, are valuable resources for further studies.

The book is recommended to those that are already familiar with $\mathsf{ZFC}$ but are unsatisfied by its presentations in standard textbooks, and its lack of clarity in rationales and motivations.