Zero to Infinity
“Zero to Infinity” is Episode 18 from Season 49 of the PBS television series NOVA. The program is 54 minutes, and the original air date was November 16, 2022. Some people are able to watch the show online, depending on which services they have access to. It is also available on DVD.
The first half of the program discusses the history of zero as a mathematical concept, as well as various number systems that existed before the base-10 system became standard. Some early number systems, like Roman numerals, do not have a symbol for zero. Both Euclid and Archimedes used number systems without a symbol for zero. A few rare books with early mathematics are displayed.
The second half begins with the concept of infinity being used for infinite limits in geometry, calculus and other fields. Hilbert's Hotel is used to introduce some properties of infinite sets. This is followed by a discussion of Georg Cantor, who played a pivotal role in the creation of set theory. Cantor established the importance of one-to-one correspondence between the members of two sets. He also developed the concept of countable and uncountable infinite sets. Any set that can be put in one-to-one correspondence with the integers is a countable infinite set. A demonstration of Cantor's diagonal argument is given to show how he proved that the set of real numbers has a larger cardinality than the set of integers. In a general sense, this accomplishment resulted in Cantor becoming the first person to provide a mathematical proof that there are different “sizes” of infinity.
People that enjoy this program might be interested in the book Infinity and the Mind by Rudy Rucker, which is available in an expanded edition from the Princeton Science Library. The book discusses the concept of infinity from a historical perspective, and provides an introduction to set theory including infinite ordinal numbers and cardinal numbers.
These are some links with additional information.
PBS | NOVA | Zero to Infinity
Amazon | NOVA: Zero to Infinity (DVD)
Rudy Rucker | Infinity and the Mind (free online browsing edition)
Princeton University Press | Infinity and the Mind
Amazon | Infinity and the Mind
Barnes and Noble | Infinity and the Mind
