**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 group**s**, 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?