Simple Principle Solves Seemingly IMPOSSIBLE Math Problems

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00:00 - Intro
01:00 - Hair twins
02:45 - Data Compression
08:34 - Different sizes of infinity

*A big thank you to my AMAZING PATRONS!*
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Creator - Jade Tan-Holmes
Script - Raphael Rosen and Jade Tan-Holmes
Cinematography and Fact checking - Simon Mackenzie
Animations - Daniel Kouts
Music - epidemic sound

[upandatom]

4:33 Whoops it's actually 22 characters, not 18! Seems like I can't count :)
14:14 Also the cardinality of the Reals is only aleph 1 if you assume the continuum hypothesis is true. Otherwise it’s 2^aleph 0. Video about that coming soon :)

[phasm]

Regarding compression, if it could really compress *any* file to 80% of its original size, then you could run it multiple times to get even smaller.

[fullfungo]

14:06 ℵ1 is not defined as the cardinality of the real numbers, it is defined as the smallest cardinality bigger than ℵ0.
The cardinality of the real numbers is denoted with 𝔠 (for continuum) or with‎ ℶ1 (beth numbers).
The question of wether bet numbers and aleph numbers are the same is called Generalised Continuum Hypothesis.

[bryanreed]

I think it's interesting that the only reason lossless data compression works at all is that the files we work with are HIGHLY atypical. The vast majority of possible bit strings are so close to purely random that a lossless compression algorithm will actually make them longer. But files we actually use are generally among the astronomically tiny fraction that have fairly reliable patterns throughout.

[davidhoward]

There are quite a few good Youtube channels covering science and mathematics, but Jade's presentation is the clearest and most logical. It's one of the few covering complex topics that I don't have to "rewind" muttering "Eh?"

[harrybarrow]

I really enjoyed this video.
I studied Maths and Physics at university — 60 years ago!
I still find the ideas I learned are useful in everyday life.
The Pigeonhole Principle is an example. Thank you. 😀

[michaelpastore]

Another unexpected place the pigeonhole principle pops up is optical illusions. A lot of them work because there are more three-dimensional scenes than there could be two-dimensional viewpoints of them, therefor some two-dimensional views must map to the same three dimensional scene.

[mrgalaxy]

The small sketches in this video were absolutely hilarious, gives such a charm to an otherwise really educational video. Great format.

[Oncampusk]

I LOVE your videos. I really like science, but am not very good at math and appreciate how you’re able to take complicated/advanced ideas and simplify it in a way that I can actually understand the basic principles. I’m not going to be an astronaut or a mathematician any time soon, but I definitely can appreciate things a bit more thanks to how you explain everything. Thank you!

[Barry]

i wish you had been my maths teacher, not till i got to about 45 did i realise just how interesting it can be. you make learning interesting that is a super power right there.

[Humdebel]

12:08 is my favourite part, i love it. The way she cross her arms is no telling

[jumpythehat]

Jade: 14:06
The Continuum Hypothesis: Am I just a joke to you?

[jurjenbos]

As a math PhD, it sort of feels obvious. But the way you tell it's story is brilliant; I'll recommend it to anyone who wants to know more about it. Thanks Jade!

[MelodiCat]

I love that you recognized implicitly in this video that the pigeonhole principle is really a statement about the cardinality of sets with surjections between each other. And I love the stuffed animal pigeons!!

To answer your question proposed: For any set A, the power set of A is the set of all possible subsets. That is,

P(A) = {B | B ⊆ A}

You can actually show (via the pigeonhole principle, essentially ;) ) that the power set of any set must always have a cardinality strictly greater than the cardinality of the original set. Here's an off the cuff proof:

Suppose A is a set. We know that for each x in A, {x} in P(A). Hence there's an injection f:A -> P(A) defined by f(x) = {x}, so |A| <= |P(A)|. Now suppose to the contrary that |A| = |P(A)|. Then there exists a bijection g:A -> P(A). Now define the set U such that

U = {x in A | x not in g(x)}

Clearly U ⊆ A, so U in P(A). Thus since g is a bijection, there must exist a y in A such that g(y) = U. Now here's the question: Is y in U? Suppose y is not in U. Then y is in U as y is not in g(y). But then if y is U, then y is in g(y), so y is not in U by definition. Thus in either case, we reach a contradiction. Thus it must be |A| ≠ |P(A)|. Since we know |A| <= |P(A)| and |A| ≠ |P(A)|, it must be |A| < |P(A)|.

Notice how we invoke the reverse pigeonhole principle here. The idea of a surjection from A to P(A) is a contradiction because the power set of a set just has all the single element sets and more. We simply have too many pigeonholes for our pigeons, so there must be more pigeonholes.

Now, because for any set A, |A| < |P(A)|, we know that |R| < |P(R)|. Thus the power set of the real numbers has a cardinality greater than aleph null.

Here's an interesting problem in mathematics that was solved in the last century: Because |A| < |P(A)| always, |N| < |P(N)|, where N is the set of natural numbers. But does there exist a set A such that |N| < |A| < |P(N)|? That is, is there an infinity between aleph null and the cardinality of the real numbers? (The power set of natural numbers has the same cardinality as the real numbers, to clarify.) Can there ba cardinality between an infinite set and its power set? The answer is that we have proved our current axioms of math (Zermelo–Fraenkel set theory with choice) cannot confirm nor deny the existence of such a set. This may sound like an unsatisfying answer, but it actually raises the question of whether the continuum hypothesis or some axiom that implies it should be added to ZFC as another axiom of math. I think that's super interesting!

[MrJohnBos]

As always, I love your videos, I'm in my 70's so I need to keep my mind sharp and I learn something new almost every time. Thank you helping keep my brain keep working.

[Mark-dcsu]

Went over this principle a couple times in Book of Proofs and *sort of* understood it. I always love when Jade comes along with a brilliant and timely explanation.

[ryanodonnell]

Nice video! But I'm afraid I have to complain that the cardinality of the reals is 2^aleph0, not aleph1. Aleph1 is (roughly) the second-smallest infinite cardinality, and whether or not this equals 2^aleph0 is the Continuum Hypothesis! :)

[annecoombes]

In the final year of my undergraduate maths course, the lecturer said he would have to invoke a principle that we had not seen before. We all groaned assuming that this meant we were going to have to either look it up or have to prove it for ourselves. However, the lecturer then went on to explain the principle, saying that if N objects are placed in M boxes and N>M, then that would imply that at least one box would have more than one item in it. This was called, he said, "Dirichlet's Box Principle". We were slightly aghast that someone had managed to get their name recorded for posterity by stating something so clearly evident. It wasn't the last time we invoked Dirichlet's Box Principle aka the pigeonhole principle.

[andrewhardy]

Quick show of hands. How many people noticed 00:21? I showed the video to my 9 year old and had to skip back several times and point it out before it clicked for him and he laughed profusely. I was surprised not everyone saw this straight away and it’s still a nice touch

[danielspivak]

The cardinality of the real numbers is not called aleph_1:
aleph numbers count infinities in increasing order of size, starting at aleph_0 being countable infinity. The continuum hypothesis, which was painstakingly proven to be independent of ZFC, is what states that the cardinality of the real numbers is equal to aleph_1, i.e. that there are no lesser uncountable infinities.