Bijective Functions and the Continuum Hypothesis

This video is largely about bijective functions. Specifically why bijective functions have inverses, why bijective functions can be used to show two sets are the same size and how the continuum hypothesis can be written as a statement about bijections.

Chapters:
0:00 - Introduction
01:02 - Definitions
03:32 - Inverses
05:28 - Cardinality
08:29 - Continuum Hypothesis

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In-video references
1. An Introduction to Set Theory (William A. R. Weiss) - Available online for free.
3. Basic Real Analysis (Anthony W. Knapp) - Available online for free
5. Cohen, P.J., 1964. The independence of the continuum hypothesis, II. Proceedings of the National Academy of Sciences of the United States of America, 51(1), p.105.

FAQ : How do you make these animations?
Animations are made in Apple Keynote which has lots of functionality for animating shapes, lines, curves and text (as well as really good LaTeX). Editing and voiceover work in DaVinci Resolve.

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Thank you so much, I hope you find the content useful.

[APDesignFXP]

I will just comment to help you with the algorithm because you’re amazing and deserve more views!

[arisweedler]

Injective surjective and bijective - useful and precise words, but not ones that have quite made it into my personal dialect of english yet. Great video!

[pablogh]

The video was excellent!!, i learned a lot. I feel like it has opened a new window or perspective for me to see mathematics. Thanks for your wonderful work.

[yeahdudex]

I don't have anything terribly meaningful to say, just wanted to say thanks for making these videos (comment for "the algorithm", as the kids say)

[jaimeduncan]

Something being undecidable does not imply that it's not true or false, but that we can't know for certain means. It could mean that we can't prove it in a finite number of steps, or it could mean that we can't know if it's true or false. It's possible that the statement is not true or false, like a pseudo-Euclidian construct without a Parallel postulate. The Parallel postulate is a degree of freedom, one can get consistent mathematic results for different choices of the postulate.

[looney]

Love this so much! I've also seen the cardinality of R written as 2^(aleph_0) and I've convinced myself that that's true but I never intuitively understood it. Would love to see you tackle the "algebra of infinities" so to speak.

[HAL-ivkd]

Question:
I'm not questioning the veracity of what's said, but I have a problem understanding the following:
Wouldn't the size of X be distinct depending of the definition of X? So, we have a general X defined as an infinite subset of R, but we don't have more information. So, if we cannot caracterize X, it wouldn't mean that we simply lack enough information about X?
Therefore, we are speaking about two completely different subsets (named X) of R and trying to force a generalization of Xsub1 and Xsub2, but Xsub1=/=Xsub2, with the two of the, being necesarrilly different by the Schröder-Bernstein Theorem presented here?

For example, all rational numbers "q" are a subset of R mith the same cardinality as N, but all x: x c (0;1), x c R (sorry for the incorrect typing, I can't find the correct symbols on my keyboard) share cardinality with R but not with N.

[rainerausdemspring]

By the way, proving the Schröder–Bernstein theorem is surprisingly difficult. Cantor stated the theorem without proof in 1887 and in 1897Bernstein and Schröder independently published proofs.
Later it was discovered that the great Richard Dedekind had found a proof in 1887 already but did not publish it.
It should be called Cantor-Dedekind theorem 😛
By the way, today Dedekind is considered - with Cantor - as one of the "inventors" of set theory.

[oida]

Would the Schröder–Bernstein theorem also work if you were to swap out injection for surjection?

[johnartzi]

This is great! But I was wondering if you will be able to make a video of how the limit is defined using epsilon delta. I am trying to learn real analysis and it’s really tough.

[digitalfroot]

Great video and easy to undertsand, thank you!! 😊

[philipoakley]

The point about the set being 'limited' to what supposed to be in it, as opposed to the content of some wider set (e.g. thinking 'all even numbers' is the set of natural number) is a mistaken perception. Part of that problem is that there is a confusion between 'counting' and the labels of the objects of the set (such as 'numbers').

The label '1', and the count of 'first element' are (can be) conceptually distinct things. This matters when explaining that the size of the 'even numbers' and the size of the whole numbers are both the same because you can count (bijectionaly) both of them.

Another example is software engineers who like to start at 'zero' (also labelled '0') as their first element of their set that's an ordered list/array. Their 'forever' is 0 to (aleph.null-1) ;-)

[johnartzi]

I thought the continuum hypothesis was disproven, I saw something online saying that

[aaronsmith]

How about the set representing all possible orderings of the natural numbers?

[chickennoodles]

So no matter how many times I rewatch this, I can’t quite tell how the bijective function g could possibly relate N to X, where N is all rational numbers and X is a subset of real numbers. Didn’t you say earlier that the set of real numbers between any two rational numbers is greater than the total number of rational numbers? so how could you possibly define a subset of real numbers as X and it be equal to N??

[xwtek]

It's not "we might never know it". We already knew it. It's true in some set theories (like in constructible universe), but false in some other set theories.

[samarthtandale]

I think it would be a little inaccurate to say undecidable as neither true nor false, isn't it?

[phpn]

Thanks for a beautiful and clear presentation. I still have a problem with Cantor's diagonality argument, which I find arbitrary. If you look at the hypothetical table represented by Cantor, I see a fallacy in the notion of rows. There is no possibility of distinct rows in the table because the decimals are infinite. All one can say about reals is that they are uncountable and as such the concept of cardinality doesn't apply there. I'd argue also that by inference, the notion of set doesn't therefore apply either to reals. Sets, I find (pardon my resorting to intuition here) must be enumerable otherwise they are ill-defined and misconstrued for computation or logic. Another, likely unrelated idea, is that reals are only meaningful between zero and one, and that number with infinite decimals are always the combination of a natural and a real.

[jaimeduncan]

The idea that the question about cardinality seems innocuous is surprising to me. I don't believe the average joe with an engineering degree will have any idea on how to even start to work on it, in particular, because is for all initine subsets of R.

[MK-]

The |A| less than or equal to |B| and vica versa being equivalent to |A|=|B| is exactly the statement that "if there exists an injective function f from A to B and an injective function g from B to A then there exists a bijection h from A to B". You can't use the conclusion of a theorem to prove the theorem 😂