What is...the axiom of choice?

Goal.
I would like to tell you a bit about my favorite theorems, ideas or concepts in mathematics and why I like them so much.

This time.
What is...the axiom of choice? Or: It works even if you don’t believe in it.

Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.

Slides.

Axiom of choice.

Books.

Russell’s paradox.

Liar paradox.

Self-reference.

Zermelo-Fraenkel and other set theories.

Axiom of induction.

Foundations of mathematics.

Reverse mathematics.

Mathematica.

Pictures used.

The axiom of choice is very popular.

[cademcmanus]

I hadn't understood AC before this because I didn't see why an element could not be selected randomly - thank you for clarifying.

[rebusd]

Thank you very very much for this explanation.

[realbart]

About Russels paradox: nobody starts describing it starts off with the hidden assumption that every statement must be true or false (the Law of the Excluded Middle). You could also adapt an alternative logic system that does not use the Law of the Excluded Middle. If you allow for statements to be undetermined, the paradox is no problem. (However, many parts of logic won't work than)

[samanosvasilias]

I love your videos, I love the format, and your humor, I hope this channel grows!

[antoniusnies-komponistpian]

Have people tried to formulate a weaker version of the axiom of choice that might work for the cases we want it to work for but doesn't produce the weird results? Or is that proven to be impossible?

[hassanalihusseini]

I like this video! Explaining very well.

I would like to see some examples of the problems with f(x+y) = f(x) + f(y) with and without the AC.

[NathanielHellerstein]

Accepting the Axiom of Choice or not is... _your choice!_

[drefplinth]

@10:29 in the video when one considers the socks to be indistinguishable (. . . sorry if what follows is a sequence of very naive questions!), is one still allowed to consider all those sets of pairs of indistinguishable socks as sets having two elements each? Or . . . should one see all of those "sets of socks" as singleton sets due to the idea of extensionality? So, in the sock example, wouldn't the axiom of choice simply be choosing the only element in each of those presumably singleton sets?

Further, would it be appropriate to consider a collection such as C = {{a}, {a, a}, {a, a, a}, {a, a, a, a}, ...} to be an "equivalence class" with its simplest element {a} taken as its representative, and should such notation (again by extensionality) collapse via C = {{a}, {a}, {a}, {a}, ...} to something like C = {{a}} (i.e., a singleton set with a singleton set as member)?

Also, is "class" in the term "equivalence class" intended to mean the same thing as "class" in the term "the Russell class", . . . or does "class" in the term "equivalence class" mean no more than what "grouping" and "collection" mean, respectively, in these artificial paraphrasings "equivalence grouping" or "equivalence collection"?

Lastly, thank you very much for producing and posting all of your videos on mathematics: they all look very intriguing!

[tim-cca]

Base on 11:03, I have a question about when we don't need AC. I am thinking whether "The Infinite Cartesian product of N is nonempty" require AC. Since we can always pick a natural number from N, say 0, we can define a choice function. So I think we don't need AC here. Am I right?

[TheMorhaGroup]

"This sentence is false" is only a paradox, when assuming a sentences claim of a truth value, makes it that value. If we are to observe the non-self referential sentence. "that claim is false" we can see. there is no reason to take the self-proposed truth value prima facie, as such a claim can be articulated, whether it is true, or not. A sentence can not determine its own truth value

[lukevideckis]

why is ZF used then if there are so many problems with it?

[NathanielHellerstein]

Koenig's Lemma is a weak version of the Axiom of Choice. I like Koenig's Lemma, and its consequent Compactness Theorem, because they validate Nonstandard Analysis, and I like infinitesimals. What disasters arise from Koenig's Lemma? Or from denying it?

[SteveThePster]

Do you have an example of a family of sets for which it is not obvious that a choice function exists? I.e. a possible counter example to the Axiom of Choice

[xbylina]

A little mistake: at 2:06 in purple box (definition of the Russel set), there should be "x NOT IN x" (instead of "x IN x")...

[Bethos-Arne]

when I heard about this axiom I thought it is obviously true. In some sense I still think it is but I am not so sure about things which follow. Perhaps this axiom should not be used. Perhaps well-ordering any set is not possible.

[apokalyptiktoxic]

I dont think that a disaster of Choice is the existente of Sets without Volume. If you haven‘t Choice your may Not be able to Proof if there is a measure or not. If you could Proof wirhout Choice that there is a measure for R^n then Choice couldnt be Independent from the other axioms

[antoniusnies-komponistpian]

Couldn't we actually say, the probability of "This sentence is false" is 1/2? Because that is not self-contradictory.

[pauselab]

It’s too useful to abandon. Pretty much all of linear algebra and field theory relies on it

[ralphinoful]

Why does the axiom of choice require a choice "function"? Why can't we just say that this choice "thing" is the existence of an arbitrary mapping? I feel like that should solve some problems.

[arigato]

Please make an intro to tensor analysis