Contributions to the Founding of the Theory of Transfinite Numbers

Georg Cantor's final and logically purified memoir on transfinite numbers was published in the late 1890s. This Dover reprint is the 1915 English translation by the mathematician Philip E. B. Jourdain; it also includes a lengthy, technically diffcult introduction by Jourdain. Contributions to the Founding of the Theory of Transfinite Numbers is not suitable as an introduction. I unwisely disregarded caution from an earlier reviewer that Cantor's work would not be appropriate for a beginner in set theory. (I thought that I was reasonably acquainted with set theory, but I do admit that I was not a math major.) The 82-page introduction by Jourdain assumes that the reader is reasonably familiar with the work of key nineteenth century mathematicians. While it is possible to skip the introduction, Jourdain's context setting is quite helpful. Cantor's transfinite numbers are so innovative and so unexpected that it almost seems as though they spring forth in a vacuum, but Jourdain shows that the earlier work of Dirichlet, Cauchy, Riemann, and Weierstrass helped point the way for Cantor. Cantor's memoir (that is, his two-part discussion of transfinite number theory) comprise the remaining 125 pages. The difficulty with Cantor's axiomatic presentation is two-fold. First, the material itself is not easy - despite Cantor's careful approach. I even bogged down for awhile on his early discussion of the exponentiation of powers and how this leads to aleph-zero. And second, much of his terminology is outdated and unfamiliar. For example, there is no mention of sets, just aggregates and parts. Another example is that Cantor speaks of reciprocal and univocal correspondence. I have yet to complete Cantor's work, but I am continuing to plod along. A recommendation: A much better starting point for readers new to transfinite numbers is a fascinating book by Mary Tiles, titled The Philosophy of Set Theory - An Historical Introduction to Cantor's Paradise. This work targets mathematics and philosophy majors, but is accessible to others.

There's nothing like reading the original. Here is the abstract theory of transfinite ordinals described by its originator, Georg Cantor.It's probably not the best introduction to set theory for a beginner. The book focuses more on ordinal numbers than on cardinals or general sets. It's not a great reference, either, since so many important results in set theory have been proven in the 100 years since Cantor. But I like this book a lot nonetheless. The exposition is beautiful -- concise, clear, and logical. It's one of the most nicely presented math books I've read.