What's so wrong with the Axiom of Choice ? 

The symbol for the empty set is not a phi, it was introduced by the Bourbaki group inspired by the Danish-Norwegian Ø (which sounds a little like the i in bird). The axiom of choice is only necessary when dealing with infinite collections of sets. If you have a finite set of sets, you can just do exactly what you said - pick an element from each set in order. Similarly, finite-dimensional vector spaces have bases without the axiom of choice. Moreover, there is an axiom of countable choice that is strictly weaker than the axiom of choice, and it is in a way more intuitive. Separable Hilbert spaces have Hilbert space bases using just the axiom of countable choice. Bases not that often used for other types of infinite-dimensional vector spaces, so that is not the main issue. A bigger concern to functional analysts may be trying to develop functional analysis without the Hahn-Banach theorem, which relies on Zorn's Lemma. However this can be done e.g. in constructive theories like Homotopic Type Theory, or Nik Weaver's constructive theories.

A of C is trivial for finite sets. Infinite sets give independence.

2:10 not exactly. If the family of sets is finite, the choice can be proven by induction. It cannot be proven for infinite families of sets. The axiom of choice is about infinity.

Thanks for your video! I have several objections to its content, but I'll just point out what seems to me the most important, because it can easily lead to a misunderstanding. As you present it, it could be interpreted as saying that the Principle of Well-Ordering (PWO) is clearly contradictory with our intuitions, because "open intervals don't have smallest elements." But this has nothing to do with the PWO. What the PWO says about the real numbers is that there is an order, a way of putting the reals one after the other, such that, according to this order, every nonempty subset of real numbers has a least element. This order will then be very different from the standard order. This cannot contradict our perception of the reals, because we don't have any idea how this order would look like and no analysis of the standard order will help us to understand this alternative way of ordering the reals.

I believe this missed the point. I consider that the axiom of choice can be better described intuitively as "generalize to infinity whatever intuitive facts we know about the finite". (This is expressed by the "for every" in the formula.) Just as a note, the axiom of choice is also surprisingly equivalent to the fact that any surjective function has a right inverse, but only in classical logic. Intuitionistically the latter is weaker.

{a,b} is very different from (a,b). {a,b}={b,a} is unordered and (a,b)≠(b,a) in general, unless a=b.

i'd like to first point out that when u said functions are tuples, u wrote a set of sets instead of a set of tuples

What you're saying at 4:14 is basically just "the usual ordering of R is not a well-ordering". The well-ordering principle states that there is _some_ ordering of R (which will look nothing like the usual one) that is a well-ordering.

Consider an infinity shelf with containing pairs of shoes. If you want one of each kind, you simply choose the set of all leftys, and you're done even for an uncountable set. That's equivalent to choosing the set of all posivie reals out of the reals (and that get's messy already when go into the complex plane as there are no "positive" imaginary numbers...). But consider a shelf filled with pairs of socks. There is not way of universally distinguing between one sock and the other in each pair and at this point you need the AoC to declare wether the task is doable for an infinite abount of socks or not.

4:14 confuses the natural order with wellorder. It would also apply to an open interval of rationals wich is trivially wellorderable as it's countable. A wellorder of any uncountable set get's quite "long" in a sense and is thus hard to imagine.

In my experience, the Barnard-Tarsky Theorem and other weird things that happen with the AoC reveal that all the counterintuitive consequences of AoC depend on the existence of pathological functions that break something in our intuitive understanding of those systems, and reveal how limited our intuitions are. In B-T, the sets are not measurable - that is, there's no way to assign to each set a volume that makes sense. Because the sets don't have a sensible way to assign volume, there is an intermediate state where the sets don't have a total volume that is well-defined, so of course you can end up with a case where you start with some volume V and end up with volume 2V. If you disallow such sets, demanding that each component set to still be measurable, then the B-T theorem collapses. Similarly with the W-O theorem; the ordering function is not the usual ordering function, and would have to be pretty gonzo in order for the W-O theorem to work. Order, after all, is additional structure on the real numbers, and a different order will have different properties. Our usual ordering of the reals is out choice out of the infinity of ordering functions available, and the AoC reveals that some of them are strange beasts indeed.

I don't think it's fair to say that "The well ordering principle is obviously false." It's just "an ordering", I see no reason at all why it should coincide with the standard ordering of the real numbers in any way.

Is the axiom of choice actually part of Zermelo-Fraenkel? To my knowledge it was actually an optional extra one which when included turns regular ZF into ZFC

Not all mathematical objects can be regarded as sets. Case in point, categories which are distinctly quite a lot larger than sets and we can talk about the category of all categories, but not really the set of all sets.

An ordering where every open set of reals has a smallest element can actually be constructed as follows without the axiom of choice: Given x and y: 1) If x and y are both irrational, x

your youtube chanel is just incredible, perfectly balanced !

the quote at the end gave me the chills lovely video mate stay healthy

Just... wow! 💜

Excelent video! Could you tell me what progam do you use to edit videos?

Amazing. Do you have any books to recommend?

Really loved it . I'd share my opinion as well in the mean time in the future.

For the open set (a,b), can't we make a bijection f: (a, b) → R₊U{0}, and then take the usual ordering of R₊U{0}, and define a well-order 𝒓 on (a,b) by taking x 𝒓 y in (a,b) if and only if f(x) ≤ f(y), then having f⁻¹(0) as the least element of (a,b) ? Like, yeah, the usual ordering for R implies that there is no least element in the set (a, b), but that doesn't mean that we can't 𝗱𝗲𝗳𝗶𝗻𝗲 a well-order on this set, as the ordering relation that we defined above is a well-order. What the well-ordering theorem states is that 𝘦𝘷𝘦𝘳𝘺 𝘴𝘦𝘵 𝘤𝘢𝘯 𝘣𝘦 𝘸𝘦𝘭𝘭-𝘰𝘳𝘥𝘦𝘳𝘦𝘥, and that's what I constructed above for the set (a,b).

It sounds like a dispute over the metaphysics of what a "set" really is. Given a set S, if you can "see" S can you also "see" its constituent members? Or when you "look" at any set does it appear atomic, with no inner structure, regardless of how complex its constituent membership graph looks? What does it really mean when we say a set "contains" another set?

For 4:00, there seems to be an easy definition to make it well-ordered, just let x ∈ (a,b), and the less-or-equal-than is defined by comparing abs(x1 - (a+b)/2)

I am for Axiom of Determinancy. Also, why did you use curly bracket for ordered pair? While it curly bracket do work if the domain and codomain are different, if the mapping is onto itself, that notation just won't work

Loved it !

can you do a video on first order logic?

Where does the cover from the video comes from ???

If you actually dig into the proof of Banach-Tarski, then the argument I find most paradoxical is not the application of choice; it is the fact that if you have the set of all sequences of cardinal steps (left, right, up, down) that end with a step up, you can step them down to get absolutely all sequences of cardinal steps.

Nice video, and cool channel.

question. there is an axiom claiming the existence of the empty set. If we discard this axiom, what does the axiom of choice look like? I.e., does the axiom of choice depend on the existence of the empty set?

I thought axioms were unprovable "premises" we just chose to assume is true (or not) and mathematics is what happens when we explore the consequences of the axioms we've chosen to use...

The Well Ordering Principle puzzled me since I began my math degree in 1978 and still does :) Because it says something exists without saying how the hell you can get it!

Personally I think the axiom of choice, similar to the continuum hypothesis is not something is necessarily true or false, but rather, it is a tool that can be used in a variety of different mathematical contexts.

The problem of finding a basis for any vector space is indeed puzzling, even in the enumerable case. Start with the field of rationals Q. The polynomials Q[X] are also enumerable and, as a Q-vector space, have the powers of X as a basis. No need of AC for that. Now consider the algebraic numbers over Q. They also form an enumerable Q-vector space. Yet we don’t have an ACTUAL privileged basis for them. We know it exists, we even know how to compute one, by enumerating algebraic numbers and checking if they are linearly independant from the previous ones. But we cannot describe such a basis in its entirety.

interesting, will defo watch again.

Now you got me really confused. How can an open interval have a maximum? But then again, how can the axiom of choice be false? It seems so intuitive.

Your definition of a function as a set is wrong. A function is not a set of sets as you put it, but a set of ordered pairs, and an ordered pair is a very specific set of sets, namely: (a,b) = {{a}, {a,b}}.

My favourite thing equivalent to the Axiom of Choice is that every fully-connected graph has a spanning tree.

No one is stoping anyone to create a whole math ignoring the axiom of choice. The trickiest part is convincing the community to work on the crooked version mathematics

A lot of people have pointed out a variety of mistakes in this video, but in my opinion the biggest error - one that I see made very often - is the idea that choice is something that we should decide is valid or not. In reality, there is no problem with the current state of affairs where we study mathematics both with and without the axiom of choice. There is no conflict here, simply two different systems both being studied.

the video is neat and short, like math videos should be, but take note of what the other commenters said about the notation for tuples, triviality of aoc for finite sets, your explanation of well-orders etc

Depends on the circumstance I'd say. There certainly are cases where it may be more useful to work in ZF instead of ZFC. I very much value that we can have a well-ordering of larger cardinals in ZFC. I did not quite get your point about the well-ordering of R contradicting our intuition of how there always is a smaller number below, how is the idea of smaller and smaller real numbers contradicted?

Love your video🤓🤓 Please do a video on topology

What’s the background music?

Set theory is not able to express all of mathematics. Firstly, you need logic before you can even talk about sets, so mathematical logic is not reducible to set theory. Secondly, there are mathematical objects that can not be expressed in terms of sets, namely proper classes. Famous examples are the class of all sets and the surreal numbers. There are other approaches to foundational mathematics like type theory and category theory which have fewer problems. Also, there are more axioms which are independent from ZF, like the continuum hypothesis. In modern math it depends on the area of research if you employ the axiom of choice or not. In analysis it is generally assumed implicitly, in finite algebra it is not (since you don’t need it). The nice thing is that all mathematical statements are of the form: If property A holds then also property B holds. So you can safely assume the axiom of choice when proving something and only if someone wants to use your result, they have to decide if they are willing to assume it themselves.

What is a vector space?

Is the category of small categories a set?

You're doing a good job, keep going, you're succeeding to convey the beauty of math

I despise the Axiom of Choice. Although Gödel proved that it is consistent with the other ZF axioms of standard set theory, his proof relies on a significant loophole in the axiom of Powerset. The axiom of Powerset asserts that every set has a powerset, but it does not clearly define the powerset of an infinite set, such as N. Consequently, the ZF axioms permit the construction of a set theory model where the "powerset" of an infinite set like N includes only the subsets with finite descriptions. This model is known as the Constructible Universe. The Constructible Universe has an extremely sparse notion of the powerset, where all sets are essentially countable. (Although the powerset of N isn't technically countable within the model, as the diagonal argument still applies.) Unsurprisingly, the Axiom of Choice holds true in the Constructible Universe, making it consistent with ZF. However, this only indicates that the ZF axioms allow for pathological models of set theory. Many people mistakenly believe that the Axiom of Choice has been proven consistent with the intuitive notion of sets and powersets, but this is not the case.

0:32 Just nitpicking here, but I think there should be a difference between ordered pair notation and set notation.

I'm team axiom of choice all the way ! just cuz somethings weird to us doesn't mean it's wrong

I attained my level of incompetence with undergraduate Maths (I did graduate, with a degree in Maths, ahem). I'll leave these questions to the Big Brains. Just realized that Mathologer has discussed this axiom. Review time.

Could someone please explain why/how there is any doubt over the Axiom of Choice being true? Surely it's not difficult to simply choose any element from each set in a collection? How is there a possible problem with that?

I am a little confused by the part about the well ordering theorem... We can prove that there is no way to order the reals, right? If the axiom of choice implies that we can order the reals, does that not mean that we have a contradiction and have disproved it? But you had also said that it was not possible to disprove the axiom of choice from the other axioms. What am I missing here?

Really Insightful ❤✌...but i just have one question or maybe more than one😁: how do we know there is nothing more fundemental than a set in math ? How come set theory explains everything but it doesn't treat mathematical logic and statements ? Aren't relations between sets are mathematical ? So there have to be a more fundemental basis for set existnce and their properties and all math in general ? ..i would be glad to discuss such questions with u ❤🍀👍

I think this is a weird case where intuition varies wildly between people. To me choice requires many other assumptions that define what choice that by the time you get to choice you have either constrained it from existing or make a long tedious specification of all the things it isn’t just to arrive at the axioms either are or arent a choice. If they are a choice, the system is predicated on choice implicitly and not in need of its own statement as it inherits it from a parent logical system. If it isnt possible, you would have ruled out all cases explicitly otherwise and the statement would thus be a corollary. The only axiom I could imagine any value in would be to define choice, specifically that it does not influence probabilities, it is constrained by them. I don’t think that’s necessary as computability constrains consciousness and that would be a higher order of logic than any statement about choice. The recursive expansion of a system of computing its own system means that it would at the very least add far more certainty than the time available to act on any non-deterministic elements. Strictly philosophically, I think the contemplation of free will is reducible to the necessity believing it exists. If it doesn’t, nothing changes because you always would come to that conclusion and if it does, you have maximized your agency. That does rely on believing maximizing agency is something that should be done, but the logic of free will inherits that judgment. In summation, I propose the only virtuous path is to type long winded rants in RU-vid comments about freewill’s irrelevance, pat yourself on the back and leave it at that.

I've met with Jerry Bona and have had some professional interaction with him, as we work in roughly the same area. He's an incredibly nice guy.

أنا أعتقد أن بديهية الاختيار تشبه المسلمة الموازية لإقليدس، هناك أنظمة متسقة بافتراض أنها صحيحة وأنظمة متسقة أخرى بافتراض أنها خاطئة ولكن لم نوجد تلك الأنظمة بعد

Perhaps I am missing some background context, but pertaining to the well-ordering principle, how does the ability to make comparisons guarantee a smallest element? If there are an infinite amount of elements, we may not have a minimum. Additionally the reals are not a countable set, and I thought this was conclusive and provable via cantor diagonalization, there is no way to number them.

The Choice Theorem is born of the inconsistency (not incompleteness) of Arithmetic. The theorem poses the undefined "plus one" of Addition against the definable prospects of Multiplication. To redeem Choice, the Theorem must include an Axiom of Hierarchal Iteration, so as to secure rigor among the applications of Choice.

I don't really think that the well-ordering theorem is 'obviously false', neither I feel that it's 'obviously true', however, as far as the open sets of reals are concerned idk what the fuss is over since it's not about ordering them necessarily by the relation of '

There is a mistake in the first order statement, it should say [... Union of A forall A in X (f(A) in A)]

So what, so not every vector space has a basis? Maybe like a zero space, or an infinite space doesn't have one? What's the closest thing to "every vector space has a basis" that you can say without the Axiom of Choice?

My understanding was that you only need the Axiom of Choice for INFINITE sets. For finite objects, things like every (finite dimensional) vector space has a basis are provable without it. The real problem I have with the Axiom of Choice is really a larger problem with inherently ill-defined and non-computable objects. Because such things generally void the warranty on pure logic itself, given that applying logic (deduction, induction, etc.) IS a form of computation. Too much of pure math has turned into questions akin to "If I put an infinitely hot burrito into an infinitely cold cooler, what temperature does it get"? As in, questions whose entire premise is logically flawed and therefore any claimed answer will be too. The Axiom of Choice or a good competitor is necessary to retain some sanity when you start playing heavily with all these infinities, and the fact you still get paradoxes is really a reflection of the fact that true reality doesn't allow infinite things.

In my mind, there is an easy way to understand axiom of choice, and when you don't need to use axiom of choice. Let X and Y be sets. Statement 1 can be derived from ZF, whilst statement 2 requires Axiom of Choice. 1. Suppose P(x,y) is a predicate such that for all x in X, P(x,y) is true for *exactly one* y in Y. Then we can construct a function f: X -> Y s.t. f(x) = y iff P(x,y). 2. (Axiom of Choice). Suppose P(x,y) is a predicate such that for all x in X, P(x,y) is true for *at least one* y in Y. Then we can construct a function f: X -> Y s.t. f(x) = y iff P(x,y). See the subtle difference? As a bonus, to also understand why we need axiom of choice,, try to work out a choice function for the real numbers. If you are given any subset in R, can you find a rule to pick an element from that subset? On the contrary, you can do this with the natural numbers (just select the minimum). Accepting the axiom necessarily means there exists things you can't construct, which some may consider problematic.

My question is if someone learn set theory could they learn all the other branches like analysis,algebra, I read this is not true because analysis came before set tehory

The Banach Tarski paradoxon sounds like something that would depend on what types of metric or topological transformations you're allowed to do. Given enough allowed transformations, we can do something close to that IRL with closed balloons (by changing their temperature). The well-ordering theorem sounds like something that would be near-impossible to disprove since it's really hard to check _all possible_ orderings of the reals. Interesting that it can be generalized to the Axiom of Choice. Zorn's Lemma is obviously true.

I don't understand what is the problem of reordering (a;b). Let's just redefine < operator for point (a+b)/2 and say that (a+b)/2 is less than any number in (a;b) except itself. Viola, we have a smallest element in open set. Problems?

After seeing that every things we define is based on assumptions called axioms, I am having a mathematics existential crisis.

This is my favourite comment section on the entire website

At 0:40 you wrote a function as the set of tuples but not ordered pairs as it is, one has {a,b}={b,a}

The interval (a,b) doesn't have a smallest element in the usual order of R. Contrary to the statement in the video, the well ordering principle doesn't contradict this fact, it only asserts that there exists on ordering of R for which there will exists a smallest element for (a,b), ... obviouslly not the same ordering scheme then the usual for R

1:34 Joke's on you. I paused the video to understand the notation, and it made sense to me.

I think that the axiom of choice is even too loose. It should be: "Given a set, it is always possible to choose ANY of ALL its elements". But is it possible to choose any e.g. real number? I have doubts, is it possible to choose a non-computable irrational number?

I've just came up with idea how to order e.g. (0.5, 1.5) real numbers interval. Let us compare numbers by absolute difference with the mean number of the interval ends, in this case with 1. Additionally, if these absolute differences of two numbers match, assume the number which is less than the mean one (in normal sence), "less" than the one greater than the mean (so e.g. 0.8 > 1.1, 0.8 < 1.2). That way the minimum element is the mean itself.

This axiom is not of "indisputable importance", because the cases where it is used in practical applications, it isn't the set-theoretic axiom of choice, but the second-order arithmetic axiom of choice, which is not controversial.

And we got our version of Euclid's 5th Axiom.

"...which side are you on?" Do I have a choice ?

One day, I'll wake up in the world where pure mathematicians realized the fallacies of transfinite induction, and all unbounded sets are necessarily the same cardinal infinitude, and Leibniz has a brand of coffee... I'm going back to sleep, now.

Wait, what? Is the well-ordering theorem equivalent to the axiom of choice? In that case both must be false?

It boils down to being a constructivist/structuralist or not. Without defining a structure of the mathematical object (in this case the set), you cannot have an algorithm for choosing an element.

I am of the side that is true, combined with the side that works. If something is obviously true, and it works, thus has practical use, then it is fine with me. You should try it in daily life too.

To me, who csn do much arithmetic abd geometry, this iscs load of silliness

Of course open intervals of reals don't have a least element when we use our normal definition of ordering. The WOP says that there is a well ordering, it doesn't necessarily have anything to do with the order we use normally. That's like saying because the negative integers clearly don't have a least element, the WOP shouldn't apply; but that's nonsense, we can just define "less than" to be what we normally call greater than, and this gives a well ordering of the negative integers. Of course no such easily definable well ordering can exist for the reals; this is exactly the point of the axiom of choice; but it shouldn't be unintuitive that one could exist.

3:16 if two theorems are equivalent, how can one be weaker than the other?

0:30 - {(a, b), (c, d), ..., (e, f)} would be correct, or in only of sets expression { {a, {a, b}}, {c, {c, d}}, ..., {e, {e, f}} }

For axiom of choice all the way 🎉

Anything self consistent is legitimate in their appropriate context.

The AoC is obviously true for finite collections of sets and perhaps also for countably infinite collections of sets. But it's not obviously true (to me) for uncountably infinite collections of sets and I believe this is where all the shenanigans come in. It can be dangerous to claim something we can't prove is obviously true. IMO the Banach-Tarski paradox proves the AoC is false in a world where you cannot magically double the volume of a sphere. So if we are basing things on what we experience in the "real physical world" then the AoC is obviously false. The most I can say is you are free to assume AoC is true and develop math that way and you are also free to assume AoC is false and develop a different math this other way. One of the main things we have learned from mathematics is we cannot trust our intuition for telling us what is True or not True. That is why we have math. Math is wonderful and beautiful because it's filled with non-obvious and non-intuitive results.

My abstract algebra professor once said: "remember that, in this class, we're believers. We believe in the axiom of choice". I agree with her

It is not true that every mathematical "object" is a set. (I am putting quotation marks because it is not clear what object even means!) e.g. the "object" which is the "collection" of all sets can't be a set because it is simply to large; it is a proper class.

Proving continuum hypothesis , proving inconsistency in ZFC , constructing ZFC from naive set specification , resolving Russell's paradox , constructing infinite number system , construct and ensure overall consistent mathematical universe and developing arithmetic system - edition 8 May 2024 DOI: 10.13140/RG.2.2.21713.75361 LicenseCC BY-NC-ND 4.0

We could use Axiom of determinism to solve all problems axioms of choice claim to be solving and not fall into paradoxes.

2:08 'Independence' is misspelled.

- The background piano music was annoying as all hell. But most information was also audio so you couldn't mute the racket... - However this was an introduction to a new field of study for me an may merit further exploration. - I am intrigued by the two equal balls constructef from one that seems to be a flaw in logicical reasoning if it is true.

Pretending I understood after 0:55

I think the Axiom of Choice needs some nerfing so that Well ordering appears only where it makes sense. Well, nothing new to Set theory. ZFC did the same for Set membership rule in Naive Set theory.

What was that at the end? The choice of axioms can in principle be false? Any how, I'm on Jerry Bona's side!

off topic doubt, set theory uses notions of logic in its development and similarly logic uses set theory, how should i think of this circularity? i've always been told that cricular reasoning is generally "bad". I'm losing sleep over this.