Cardinality of the Continuum

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:
Roads to Infinity: The Mathematics of Truth and Proof by John C. Stillwell

Bright Ideas by Purple Planet Music
Confusion by C418
Kitten by C418

Animations were made by Manim, an open-source python-based animation program by 3Blue1Brown.

This video was submitted to 3Blue1Brown's SoME2 (Summer of Math Exposition 2).

[ehxolotl]

Small issue at 19:38, 1 should translate to 0, not 00.

[aiaian_aaa]

This video is incredible. Actually it’s THE best video which talk about infinite sets. The pace and music be just perfect and when it comes to the end, it literally teaches us more than just math. Also the words you used are very friendly for a English learner like me. We need more math videos like this!

[scottcampbell]

The cardinality of the set of useful Youtube math videos has increased by 1.

[harrytaylor]

I adore the 3B1B-like style used here. The way it was used in the infinite-primes proof was very smooth

[marcoottina]

17:00 The musical choice is fantastic! It truly whisper the idea of "there's some powerful concept here to be grasped, but it's doomed indeed".
Genius!
Also, the part immediately preceding it has got a VERY inspiring music, which lift the spirit to the idea of "this concept is very powerful, harnessing the power and patience of infinity to gain the ability to be exact"! (17:00 _unless_ ... *there's a hole** )

also, the ending speech and the "almost pun" is ... delightful!
I love it!

[sleepyhaad]

please post more frequently, I genuinely loved this video.

[chasg]

Wonderful presentation! A small detail, the background noise is distracting and takes away from the experience. Regardless, please make more!

[evgenysmirnov]

The conclusion part — brilliant execution: the script, the music, the eloquence — this is vsauce quality, good job!

[susmitislam]

Very well made! If you can, please continue making similar videos. 💯

[jonahansen]

Dude - please can the music, or at least tone it down. It distracts my right brain. Not to disparage this video, it's fantastic. Thanks! Shoot - I didn't read the comments before I put in mine; someone else also mentioned this.

[cecil]

This video deserves much more views than it has right now.

[ouvie]

You explain it well and you're quite underrated. Keep it up!

[alegian]

For some reason this is the first time Ive come across the double-representation edge case, really interesting 👍

[beautyofmath]

This is a wonderful video about infinite sets, u explained clearly and I learned a lot, thank you for this.

[supergeniodelmale]

The binary coding step was simply incredible! Keep it up!

[inciaradible]

I remember first learning about this in university; it honestly makes infinity feel like something you can grasp. The bijection argument for sets being the same size is just so intuitive.

[gooball]

Awesome video! The pacing felt right and still you managed to delve into some very interesting insights in a short amount of time. Top notch math education!

[TheoriesofEverything]

Great video man. I shared it in a recent video on "math channel recommendations." (albeit only the description) By the way, do consider submitting to the contest on the channel. Email me for details if you're unsure. Continue the enlightening work.

[ashleemeow]

For the uncountability of the unit interval proof I would choose a slightly different example of a number not in that set. It's possible that you end up with a number that ends in a string of zeroes using this method and since such numbers don't have unique decimal representations it's also possible this number is already in your set. If you change the numbers you use from 0 to 1 into 1 and 2 in the algorithm you avoid this issue without having to make further assumptions or proofs :)

[kcthomas]

I think a much better approach is to point out 0.1000... and 0.0111... (in binary) are in fact equal in value, in the exact same way that 1.000... and 0.999... (in base 10) have equal value (both are equal to 1).