Customer Reviews

Set Theory and Its Philosophy: A Critical Introduction

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I believe one has to have some familiarity with logic and set theory in order to fully appreciate this wonderful book. Granting that, reading it was the first time I have ever read a mathematics book that I could hardly put down, it was so fascinating.

When I was an undergraduate, a course in naive set theory (similar in content to Halmos' classic) persuaded me to become a mathematician. But when I asked my instructor to precisely define what a 'property' of a set was, a notion that was used in the Axiom of Separation, he evaded the question as too philosophical. Much later, when I studied mathematical logic, I found a precise definition.

Michael Potter does not seem to evade any philosophical questions about set theory. The answers he proposes are given from various points of view so the reader can clearly see the differences and possibly choose the one most congenial: platonism (internal, uncritical, limiting case), constructivism, formalism (pure, postulational). I couldn't pin down exactly what is Potter's point of view except that he is not a strict formalist or a strict constructivist or an uncritical platonist.

His development of the purely mathematical part of set theory is very elegant, especially his axiomatization of the levels of the set theoretical hierarchy. Unlike most strictly mathematical texts, Potter explains why, at each major stage, he is doing what he is doing. In three appendices he also contrasts his approach with the traditional ones. I felt he did not give enough credit to the simplicity and elegance of NBG theory, so well presented in Mendelson's classic text; he is averse to introducing classes as well as sets.

His treatment is replete with fascinating history. He does not hesitate to discuss advanced results which he cannot prove in a treatment at this level, and he provides ample references if the reader is interested in pursuing them.

I am still puzzled by the nature of second order logic, which he says "decides" the continuum hypothesis, which is an undecidable statement in first order logic. I wish he had explained that more.

This is a book that I intend to re-read and to discuss with colleagues who are expert in the field. Very highly recommended.

I full concur with Greenberg's review. Assimilating Potter's book is also much easier if one has had a prior introduction to mathematical logic and axiomatic set theory.

Potter sets out an axiomatic set theory he calls ZU, whose axioms are: there is a ground level of sets, every level has a successor level, Infinity, and Reflection (a schema). These axioms are a perspicuous embodiment of the iterative conception of sets and the related hierarchical ontology. Potter then shows that these axioms achieve, in a fairly relaxed way, all we would want these axioms to achieve. This theory should be given an important place at the high table of foundational mathematics.

Set theory is inherently philosophical because its true subject matter is patterns in the human mind and human sensory experience (in this respect, I concur with Lakoff and Nunez). Potter is a bracing philosophical read, but be aware that there is a good deal more to the philosophy of set theory and foundational math than he lets on. His ample bibliography nicely shows the way to more reading in this vein.

Some intellectual history. In the 1960s, the mighty Dana Scott began working on a new axiomatization of set theory, grounded in type theory and the iterative conception. This work culminated in a talk he gave at a 1971 conference, whose proceedings were published in 1974. Scott was also supposed to be working on a monograph on set theory with Montague, who died in 1971, and Tarski, who died in 1983. The monograph will never appear, and Scott never fleshed out the intriguing proposals he published in 1974. Potter's book is the belated bloom of Scott set theory.

Greenberg is right about Mendelson's intro to NBG; it is a good introduction to the mechanics of axiomatic set theory, independently of Potter's book. Potter is also not as well disposed to Quinian set theory as I am.

I am puzzled by Potter's claim that Skolem arithmetic (just multiplication over the naturals) is finitely axiomatizable. Cegielski (1981) firmly asserts otherwise.

This is a superb book, but it has a very specific audience. It is a careful, systematic, investigation of the extent to which the methods of set theory can be used to address philosophical questions. So the audience needs to both be comfortable with the formal presentation of mathematical theories, and to know the issues in the philosophy of mathematics. If you lack the philosophical part, you'll wonder why Potter doesn't just use ZF, and why he keeps being drawn off into various topics along the way. If you lack the mathematical part, you'll find the book hard to understand, although it is extremely systematic. (If you don't know what ZF is, for example, I'd advise starting with some other book.)

Having said that, Potter goes out of his way to present matters clearly and explicitly. Readers who don't exactly fit the audience will learn an enormous amount from this book. Moreover, it is so clear and authoritative, and covers so much ground, that it deserves to be in the canon. It ought to displace Quine's Set Theory and its Logic, for example.

ZU is Potter's set theory (76). It is spare, and very powerful. I believe Potter is trying to capture as much as he can of Frege's original view of sets as logical objects, although he doesn't say this. ZU allows flocks of doves and packs of wolves to be sets, just as it intuitively ought to, but it can also capture the real and transfinite numbers. The book divides into four parts. First, there is the presentation of ZU and its properties. Then we get the usual development of the real numbers. The third section deals with ordinals and cardinals, and a fourth section the axiom of choice and the continuum hypothesis.

What sets the book apart, though, is its constant return to the history of its subject and the philosophical issues that have been embroiled in it up to the present. You can look through the book at any issue that interests you - Russell's paradox, non-standard analysis, whether there is some deeper notion of a collection underlying set-theory - and Potter always gives a clear explanation and has something interesting to say about it. With graduate students of sufficient ability, the book would make for a really worthwhile graduate level course in philosophy.

When I'd finished reading it, I wanted to read it again.

I have dropped the book after in(digesting) chapter 3 on hierarchies which amount, via Potter's cryptic style, to a hierarchy of horrors... One has almost to rewrite this chapter i.e. annotate every paragraph...

I finally turned to Suppes's "Axiomatic set theory", which is a little better, although... (see my review).

Why is it that Set Theory seems to count very few true didactically-talented authors of the caliber of Gauss, Hardy, Apostol, Tarski, Coxeter, Courant, Kline, Greenberg (though I don't share his enthousiasm for Potter's book), Russell (except for the Principia, the crowning example of unreadable masterpiece), Quine, Smullyan... ?

I might return to that book, one day, and try once more to overcome its repulsive aspects.