Zermelo's Axiom of Choice: Its Origins, Development, and Influence

by Gregory H. Moore

This book chronicles the work of mathematician Ernst Zermelo (1871–1953) and his development of set theory's crucial principle, the axiom of choice. It covers the axiom's formulation during the early 20th century, the controversy it engendered, and its current central place in set theory and mathematical logic. 1982 edition.

Genres: Mathematics, Logic, Own

448 pages, Paperback

First published January 1, 1982

Reviews

[Moses · Rating 5 out of 5]

More often than not I fail to realize that history is a continuous process, and this is especially true considering the history of mathematics and the foundational crisis at the turn of the 20th century. It is very interesting to see how our modern idea of ‘set’ and the unintuitive nature of Choice came about from Cantor and Zermelo despite the seemingly unreconcilable paradoxes of Russel and Burali-Forti, and I think Moore does a fantastic job of balancing rigor with historical context. Overall, one of the most readable and engaging surveys of mathematics that I feel is accessible to the layman with some concentrated effort.

[Neal Alexander · Rating 4 out of 5]

The axiom of choice says that for any, possibly infinite, collection of sets, it’s possible to choose one element from each set. It may not sound like a big deal. On the other hand, paraphrasing Bertrand Russell: to pick one shoe from an infinite series of pairs, you don’t need the axiom of choice, but for socks you do.

There’s a joke which goes: “The axiom of choice if obviously true, the well-ordering theorem is obviously false, and who knows about Zorn’s lemma.” The point being that the three statements are mathematically equivalent, yet not equally intuitive. In fact, Zermelo made the axiom of choice explicit in order to prove the well-ordering theorem in 1904, not yet having realised that the two are equivalent.

Many philosophers and mathematicians objected to the axiom of choice because it says that an infinite number of arbitrary choices can be made, without specifying how. Yet even some of the axiom’s critics, including Russell, had implicitly used it in their work. Zermelo brought to the surface disagreements which had been simmering for decades, and the controversy continued for decades more. For example: Tarski proved that the axiom of choice is equivalent to the statement that every infinite set S has a bijection with the Cartesian product SxS. He sent his paper to the Comptes Rendus de l'Académie des Sciences in Paris. One reviewer rejected it on the grounds that an implication between two well-known results is not novel. The other rejected it on the grounds that an implication between two false propositions is of no interest.

One might think that the axiom of choice was transiently controversial in the same way as negative or imaginary numbers had been in the past. However, unlike those other examples, some mathematicians still reject the axiom of choice. In particular, the constructivists do so, for the reason already mentioned, i.e. they consider that entities do not exist, even Platonically, as long as they have not been defined explicitly. Others reject the axiom because it leads to entities which seem to speak to physical reality but don’t have physical properties (in particular, unmeasurable sets). On the other hand, as Leonard Wapner said in his book on the Banach-Tarski theorem, the axiom of choice is an axiom of mathematics, not of physics. Overall, though, nowadays mathematics is in a period of Kuhn might’ve called “normal science”. As Moore says on his first page, “mathematics has oscillated between studying its assumptions and studying the objects about which those assumptions were made” and for now we are in one of the latter periods.

Moore is a mathematician but the book has a strong historical sensibility. A good example is the description of the development of logic and set theory in Poland after World War 1 (on Poland’s renewed independence). This was because the relatively few mathematicians realised that if they were going to have any kind of critical mass then then had to focus on specific fields and so, for example, they founded a journal with a specific focus on the foundations of mathematics.

One unexpected character is John Tukey, known to me for statistical methods but whose PhD was on topology, and is mentioned here in relation to another result in set theory that’s equivalent to the axiom of choice.

For anyone who has sufficient mathematical background to follow the technical gist, the author has written a very readable account of a period in mathematics when many were disputing the fundamentals of their field.

[Cynthia · Rating 4 out of 5]

This is difficult going, as you might have guessed from the title. Worthwhile so far, but (at least in the long introduction) like reading a math paper every sentence & every paragraph.

I'd recommend this to Todd - and nobody else that I'm aware of, unless you have PhD-level training in metamathematics.

[Jim · Rating 5 out of 5]

A superb summary of the history and influence of the Axiom of Choice on mathematics, and why mathematicians had so much difficulty accepting it (something I never really understood).