The Mystery of the ALEPH, Mathematics, the Kabbalah, and the Search for Infinity. by Amir D. Aczel.
I picked up the book on one of Barnes & Nobel’s elCheapo bargain tables for only $7.99. It was a real buy as I thourghly enjoyed the entire read. The central theme was infinity and it’s place in Mathematics History from Greek times through the present. Brief descriptions of the contributions of many important mathematicians are woven together with emphasis on basic principles, and especially the contributions and tribulations of Georg Cantor. The author makes a good case that “the intensity of actual infinity” caused Godel, Cantor and perhaps others to descend “into madness from the intense introspection that studying the continuum hypothesis entailed.” I had never understood that there were multiple orders of infinity.
I liked the review of classifying numbers (and their understanding through history): integers, reals, irrationals, etc. I was particularly taken with the ideas on pages 111-113 where he talks about the countablility of integers and rationals. Cantor’s Diagonalization Proof of the Denumerability of Rational numbers. For the hell of it entered the “infinite” “table” of rational numbers on page 112 to an Excel spreadsheet. It helped me see the relations between some fractions and “repeating decimal” numbers… (I’ve always been taken with the fact that such very common fractions (eg. 1/7 (.285714….) have a small number of digits that repeat for ever) )
I will be amazed if somebody both reads this esoteric Blog entry AND takes the trouble to look at this spreadsheet. Ah, the journey of it all….