Set Theory and Its Philosophy: A Critical Introduction

Rating:
Summary: Unique blending of mathematics and philosophy
Review: 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.