Extended Course Description. Course materials: One book (see below); other materials will be made available free in PDF form or with links to online sources.

  • Book: The Mystery of the Aleph, by Amir Aczel (available through bookfinder.com as low as $11.75 new or $3.49 used). A readable introduction to forms of infinity and to originators, especially Georg Cantor (1845-1918), of the underlying theory.
  • PDFs of two sections from PZ-authored textbooks, touching on infinity and cardinality.
  • Short readings from general and literary sources, including “The Library of Babel,” by Jorge Luis Borges and an article on Zeno’s paradoxes from Scientific American; in PDF form.
  • Three weekly “homework assignments”: wholly optional but ideally revealing problems and puzzles, more to be pondered than completed.

Notes: Please read and/or work on these selections before class so that we can meaningfully discuss them. We will spend some class time in breakout groups, for which it will be especially helpful to have pondered some of the “homework” problems; see definition above.

Class 1: The basics: big, small, finite, infinite.

Reading: Aczel book, through page 24.

Class topics: Getting started. Euclid’s proof and the infinitude of primes. Basic examples and a first look at Hilbert’s Hotel. 

Live questions: Does infinity exist? If so, what is it? Is the universe infinite? Is infinity a number? If so, can we do arithmetic? Can anything be infinitely small? 

Sets and subsets. Finite sets (“easy”) and infinite sets (“hard”), with key examples. The pigeonhole principle. Cardinality as the “right” measure of size for sets. The Cantor-Schröder-Bernstein theorem: a big name for a basic but subtle result.

Class 2: Paradoxes, ancient and “older” history.

Reading:More of Aczel book. Zeno’s paradoxes. Start PZ textbook sections on cardinality.

Class topics: Check-in on “homework,” including return to Hilbert’s Hotel. More on cardinality. Positive integers equinumerous with all integers. Zeno and friends. Infinite sums and their arrangements.

Live questions: Can infinitely many positive numbers have a finite sum? Is movement possible? What does this have to do with infinity?

Class 3: Modern history, “big” and “small” infinity.

Reading: Aczel book through page 190; skimming OK.

Class topics: Check-in on “homework.” Beyond rationality. The square root of 2 is not rational, but it exists. Why it exists. A glance at completeness of the real numbers, and why it matters. Cantor’s diagonal proof that real numbers are uncountable. Limits. Derivatives, integrals, and infinitesimals.

Live questions: Which sets are “countable?” Are the rational numbers countable? The reals? Why? What does this have to do with the calculus?

Class 4: To infinity and beyond.

Reading:Finish all readings, including Borges (only 9 pages). Investigate on your own, if interested: Ideas of infinity in the bible. Augustine on God. Anselm on God. Berkeley on “departed quantities.” More on the Kabbalah.

Class topics: Check-in on “homework.” Discuss optional topics. Infinity and divinity. Cantor, Russell, alephs, and the continuum hypothesis. The axiom of choice and Zorn’s lemma (that’s another Zorn). ZF and ZFC theories: Pro-choice or anti-choice?

Live questions: Do infinities “really” exist? How “big” is the set of real numbers? How many books are in Borges’s library? Is the universe infinite? If not, will it “run out” eventually?