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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: primes.utm.edu/notes/proofs/i...
Proof that the cardinality of the unit interval is the same as the cardinality of the reals: • The Cardinality of an ...
Roads to Infinity: The Mathematics of Truth and Proof by John C. Stillwell
Wikipedia article about Cantor's Diagonal Argument: en.wikipedia.org/wiki/Cantor%...
How to Count Past Infinity by Vsauce: • How To Count Past Infi...
Image of Euclid: cdn.britannica.com/46/8446-05...
Music: www.purple-planet.com
Bright Ideas by Purple Planet Music
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.
github.com/3b1b/manim
This video was submitted to 3Blue1Brown's SoME2 (Summer of Math Exposition 2).
summerofmathexposition.substa...