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Proofs of the Cantor–Bernstein Theorem: A Mathematical Excursion

Arie Hinkis
Publication Date: 
Number of Pages: 
Science Networks Historical Studies 45
[Reviewed by
Michael Berg
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I have had a great deal of practice over the last quarter century teaching the all-but-ubiquitous transitional course for mathematics majors and fellow travelers, taking students from lower division courses, where methods reign supreme, to upper division courses, where proofs are — or should be — the order of the day (every day). Leaving aside any discussion (or lamentation) of how times have changed, it is something of an elixir to be able to introduce a measurable amount of set theory into the game, despite its having less resonance nowadays than was the case when I was a rookie prof. The upside of this is that the more gifted kids tend to respond very positively to the exciting things they see, e.g. the construction of the natural numbers in the style of Zermelo and Fraenkel, and the marvelous play of different levels of infinity. It is always great fun to show them that the set of fractions looks big, but isn’t (i.e. it’s only countable), while the reals at first glance look no worse than the fractions (consider the denseness business, for instance), but they are — much worse, in fact: you can’t list ‘em, as Cantor himself so brilliantly demonstrated via the so-called diagonal argument that fails.

What’s lurking in the shadows is the adder’s nest of questions about not just the meaning of mathematical infinity, but such noxious themes as intermediate infinities, (with the continuum hypothesis darting out to bite you in the ankle). Then, as the dust begins to settle, a natural hierarchy emerges, based in large part on the fact that a set is never equinumerous with its power set. For bemused sophomores that’s something quasi-familiar at least: if we are dealing with a finite set, this result is true and clear to even meanest intelligence, given that n is always strictly less than 2n.

At that point we have effectively reached the result the book under review is devoted to, namely, the theorem of Cantor-Schröder-Bernstein. A propos, given the title of the book, one is moved to wonder: whatever happened to Schröder? Well, here’s what Hinkis says on the first page of the book’s Introduction:

In [Alfred North] Whitehead’s 1902 paper, in Section III, written by [Bertrand] Russell, [the] C[antor] B[ernstein] T[heorem] is referenced as “the Bernstein’s and Schröder’s theorem” … after the two mathematicians who first published proofs for the theorem, both in 1898. In 1907 … [Philip] Jourdain suggested that the name be “the Schröder-Bernstein theorem.” He thought that Cantor’s name should not be included … because Cantor did not provide a proof of it, a mistake that has often been repeated. As it turned out, the name of Schröder should have been omitted ... because it was found that his proof is erroneous (or rather, as will be demonstrated later, senseless).

The proper statement of CBT is not what I alluded to above: that result one gets as a corollary of CBT. The theorem of Cantor and Bernstein itself states that if a set M is equinumerous with a subset N' of a set N, and if N is in turn equinumerous with a subset M' of M, then M and N are equinumerous. This, though nothing if not plausible, is representative of what transpires in set theory, a discipline not unlike elementary number theory in the sense that assertions which seem eminently reasonable if not close to obvious require an incommensurate amount of effort to prove.

The book under review, a paean to CBT, is in a sense an illustration of this characteristic of set theory, even as it is a scholarly and history-saturated meditation on the class of results that accrue around it. However, so as not to flirt with disingenuousness, I must stress that it is also characteristic of set theory, particularly in an area such as the study of infinite sets per se, that the requisite methodology is extremely arcane and often quite austere.

That having been said, Hinkis’ subtitling of his book as “a mathematical excursion” promises to mitigate the threatened cataract of mathematical logic and, to be sure, the book is chock-full of history, historical analysis, discussions of major (as well as minor) Briefwechseln (e.g., Cantor-Dedekind, with a great deal of other arcana added; consider, for example, the following passage on p. 75: “… we wonder if the fact that Cantor got married and had children while Dedekind remained a bachelor living with his unmarried sister is not a factor to count [regarding the ups and downs of the relationship between the two scholars] … [or] their differences in wealth … ancestry … religion and religiousness, political views, extra-mathematical interests?”). The description “excursion” is obviously on target.

Qua mathematics as such, we encounter no fewer than five parts to this 400 page opus, in order: “Cantor and Dedekind,” “The Early Proofs [of CBT],” “Under the Logicist Sky” (e.g., Russell, Jourdain, Poincaré [!], Peano, Zermelo, often with their own proofs of CBT), “At the Polish School” (Sierpinski, Banach, Kuratowski, Tarski, and others), and “Other Ends and Beginnings.” Marvelously, the book’s final two chapters deal with CBT and intuitionism (with the usual Dutch suspects coming into the picture: Brouwer, van Dalen, Troelstra) and CBT in (yes!) category theory — the very last section is particularly tantalizing: “On possible origins of the commutative diagram.”

So, all in all, Hinkis’ Proofs of the Cantor-Bernstein Theorem is a true labor of love by some one who not only knows an awe-inspiring amount about this beautiful result and its many ancillae and addenda, but is equally keen on explicating the mathematical and cultural history surrounding this part of set theory and logic. To boot, the men who populate the pages of the book are often very familiar to all of us, and the information and analysis Hinkis presents nicely (and richly) supplements what we already know about them. It’s a wonderful book — for the right reader: there’s the obvious caveat that it is truly all about set theory and logic, which is not every one’s cup of tea.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.