Cantor's Infinity Paradox | Set Theory

Cantor sets and the nature of infinity in set theory.

Hi! I'm Jade. Subscribe to Up and Atom for new physics, math and computer science videos every two weeks!

Visit the Up and Atom Store

*Follow me* @upndatom

A big thank you to my AMAZING PATRONS!
Robert Guzman, Skully, Paul Kendra, Ofer Mustigman, Daeil Kim, Harsh Tank, Alan McNea, Daniel Tan-Holmes, Simon Mackenzie, Andrew Styles, Dag-Erling Smørgrav, Chris Flynn, Andrew Pann, Anne Tan, Joe Court, John Satchell, John Shioli, Adam Thornton, Ayan Doss, Marc Watkins, Sung-Ho Lee, Todd Loreman, Susan Jones, Jareth Arnold, Fernando Takeshi Sato, Armin Quast, Xiao Fan, Simon Barker, Shawn Patrick James Kirby, Simon Tobar, Drew Davis, Rob Harris, Dennis Haupt, David M., Ammaar Esmailjee, M.H. Beales, Doug Cowles, Stephen Veitch, Renato Pereira, Simon Dargaville, Noah McCann and Magesh.

If you'd like to consider supporting Up and Atom, head over to my Patreon page :)

For a one time donation, head over to my PayPal :)

*Music*

*Sources*
The Annotated Turing - Charles Petzold

Introducing Infinity, A Graphic Guide - Brian Clegg

[palrob]

I love how goofy your drawings are. They look very endearing.

[dubber]

mathematician comes to a bar, asking for a 1 litre of beer and then asking half of it, and half of it, and then half of it. The bartenders then said, "You should know your limit".

[giwrgos_kakep]

"You can't really count all the natural numbers, you die at about 3 billion" - instant subscribe

[legoharry]

"My infinity is bigger than your infinity." - Thanos

[hexenex]

Please, don't apologize for the length of the video. It is extremely fascinating and informative. And, infinite, of course!

[SahilBansal]

In the proof at 04:22, in the first line, it must be suppose a/b = sqrt(2).

Ah great video! It reminds me the first years of my engineering with calculus and algebra and their demonstrations. I could find the beauty of some demonstrations, and some other nasty ones... There were some challenges of course, but at the end finding the order in the chaos was part of the joy too. Thanks for the explanation, that I'm sure has hard work behind.

[AlexKnauth]

Wait the “proof by continuum” thing at 7:43 : why doesn’t that also imply that the rational numbers are non-enumerable?

If the rational numbers are enumerable, and this proof technique seems to contradict that, is the proof technique invalid?

Or, does it not actually apply to the rationals because of something I’m not seeing?

[theultimatereductionist]

Order density of a set (between any two x & z in the set there exists a y in the set such that x < y < z) implies automatically an infinite set. But, you need the additional & independent axiom of the least upper bound property to make the set uncountable. Both Q & R have order density but only R has the LUB property.

[therealEmpyre]

I see an issue with the first proof you showed for real numbers not being enumerable. You said they are not enumerable because there are infinitely many other real numbers between any two real numbers, no matter how close they are. This is true, but the same is true for rational numbers, which have been proven to be enumerable, so that is not a proof that real numbers are non-enumerable.
The diagonal proof is convincing, though.

[rish]

Another lovely video. I love how you show a proof by induction without getting into the guts of how to prove by induction! Brilliant.

[prbroussard]

Yeah! Always fun to hear about Aleph null and Aleph one

[russellcannon]

Wow. Wow! WOW! Very well done. Thank you so much for making this video. I have known for a long time that some infinities are larger than others, but whenever I tried to explain this to my skeptical friends, I would fail miserably. The explanation you provide here is so elegant and succinct that all I need do is recommend this video to my friends as THE proof. Thank you so very much. A true measure of genius is not in what one knows but in the ability one possesses to successfully explain a complex concept in a way that can be apprehended by those who struggle with it. You are a genius. Vey well done. Cheers, Russ

[SpencerTwiddy]

Cool vid as always👍🏼 here's a quick video topic suggestion: the Anthropic Principle. It might be a bit philosophical but I hear about it a lot from leading physicists, and think you would probably have an interesting and unique way of explaining it. Thanks!

[coreybray]

We note first that the squares of N are contained as a proper subset within the set N. So, if N mapps to the squares of N, and the squares of N maps back to the squares of N in N, then the set N must have a Cardinality that is larger than its own Cardinality, a clear contradiction. This arises from the fact that we did not exhaust N in our second mapping, but only exhausted a subset of N.

In fact, if we use the standard definition for an infinite set, we run into some serious and blatantly overlooked problems.

For all m in Z^+, there exists an element m+1 in Z^+, such that m < m+1 in Z^+. This set Z^+ is endless by construction, because for every m in Z^+ we know we always have one larger value m+1. But, at no time do we everfind, when building Z^+ that the transition from m to m+1 is ever a leap from the finite to the infinite. There is always a finite distance of +1 that exists between all values m and m+1 in Z^+. Consequently, the set is both endless and finite everywhere, not endless and infinite. Anyone doubting this result is encouraged to identify the value for m, such that m+1 is no longer finite.

Also, notice that mathematicians argue that infinity = infinity + 1 = infinity + 2 and so on, but if all these steps equal infinity, then the very idea of magnitude, as well as the fundamental theorem of arithmetic, breaks down at infinity. But, the concept of magnitude and the fundamental theorem of arithmetic never breaks down in the case of m < m+1, because m and m+1 cannot possibly be represented as the same unique product of primes to their powers as with the earlier case where all the above forms of infinity equal each other. So, Z^+ is not an infinite set, because all values of m and m+1 in Z^+, such that m => 2, will never have the same magnitude, nor the same unique prime factorization.

[linkon_]

Thank you for explaining such a complex topic so easily. It takes some time to grasp the content of the video, but still explained effectively. 🙌👏👍

[thomaswright]

Props on the Charles Petzold quote. 'CODE' is in my top 5 most important books I've ever read.

[equesdeventusoccasus]

Excellent work. It feels like this could be the start to a new series you could post along side your other videos as time permits. I'd be happy if it was.

[impowerzone]

Your expiation of the topic was amazing. I was struggling with some of the concepts while learning discrete mathematics for my data science course but you made it fun and easy to understand, please keep going your work is useful in many ways :) thanks for making the video.

[dcryan]

Loving your videos! 👏 You really know how to break a topic down into bite size pieces. You‘re able to take comprehensive topics and make them more approachable and less daunting.