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