Cardinality of the Continuum

Video Description

What is infinity? Can there be different sizes of infinity? Surprisingly, the answer is yes. In fact, there are many different ways to make bigger infinite sets. In this video, a few different sets of infinities will be explored, including their surprising differences and even more surprising similarities. 0:00 - Euclid's Proof of Infinite Primes 1:55 - Bigger Infinities? 2:27 - Set Theory and Bijections 5:12 - No Countable Difference Principle 6:13 - Power Set of the Naturals 8:12 - Euclid's Proof and the Power Set 9:20 - Cardinality of the Reals 10:57 - Cardinality of Positive Integer Functions 13:29 - Are these Cardinalities the Same? 14:11 - Binary Notation 17:44 - Real Numbers and the Power Set 19:19 - Functions and the Power Set 20:56 - Conclusion Additional Resources: Euclid's Proof of Infinite Primes: https://primes.utm.edu/notes/proofs/infinite/euclids.html Proof that the cardinality of the unit interval is the same as the cardinality of the reals: https://youtu.be/y7Jnf-REBEM Roads to Infinity: The Mathematics of Truth and Proof by John C. Stillwell Wikipedia article about Cantor's Diagonal Argument: https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument How to Count Past Infinity by Vsauce: https://youtu.be/SrU9YDoXE88 Image of Euclid: https://cdn.britannica.com/46/8446-050-BC92B998/Euclid-woodcut-1584.jpg Music: https://www.purple-planet.com Bright Ideas by Purple Planet Music https://c418.bandcamp.com/album/mixes Confusion by C418 Kitten by C418 Animations were made by Manim, an open-source python-based animation program by 3Blue1Brown. https://github.com/3b1b/manim This video was submitted to 3Blue1Brown's SoME2 (Summer of Math Exposition 2). https://summerofmathexposition.substack.com/p/the-summer-of-math-exposition-is?s=r