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- เผยแพร่เมื่อ 14 ต.ค. 2024
- One of the Zermelo- Fraenkel axioms, called axiom of choice, is remarkably controversial. It links to linear algebra and several paradoxes- find out what is so strange about it !
(00:22) - Math objects as sets
(00:54) - What axioms we use ?
(01:30) - Understanding axiom of choice
(03:27) - Axiom leads to paradoxes
#Axiom
#ZFC
#Settheory
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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.
I came to this video cuz I had an argument with my friend over ZFC icarus set and absolute infinity, he said since ZFC icarus set resolves cantor's paradox along with many, ZFC icarus set is actually bigger than absolute infinity itself, but from what i read absolute infinite should be the epitome of all the numbers, it breaks the notion of size itself, but I can't find anything on the net for confirmation, so I decided to study into ZFC and came here, but I'm really new to infinite series and cardinality, so would you pls clear my doubt on which is bigger, ZFC icarus or absolute infinite? Also if you can give reasonings on how, that would be really helpful, so do you know something on this topic? Would really help me out :)
@@gauravdoley1527 I don't really know anything about the Icarus sets, sorry!
@@TheMaginor it's alright mate dont sweat it, thanks for replying either way!!
@Gaurav Doley thanks, that's helpful
My first thought
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.
Yeah, that's the point of being contradictory; because such an order would imply that, no matter how weird or diverse your subset S of the reals R, there is always going to be some smalest element... if you think carefully about this that maybe could imply that R is a countable infinity set
@@tonaxysam No, there is no contradiction there either. There are all sorts of well ordered sets, some of them are finite, some of them are countably infinite, and some of them are uncountably infinite. What the PWO says about the real numbers is that there is a well ordered set with as many elements as real numbers there are.
@@jgilferez Yeah, but if some well ordered set with the same amount of elements as the reals exists; there must be an isomorphism with some order that preserves the order of that set...
@@tonaxysam For infinite sets you can get two sets with the same cardinality but different well order structures
For example, consider the order
1 < 2 < 3 < ... < w
where I've added a single element w that is bigger than every natural number. It is a well ordered set of cardinality the same as the natural numbers. However, it's not order equivalent to the natural numbers because the natural numbers don't have a maximal element, but my new set has the maximal element w
@@StewartMcGinnis @Stewart McGinnis Oh, so I guess that if a ordered the real number like:
-1 < r < 1 for all the r on the Reals, with r not equal to -1 and r not equal to 1
That order would be a well order?
Well that would actually happen right? I mean, every single not-empty subset of R will have a least element, right? But like, the least element has to be on the set?
A of C is trivial for finite sets. Infinite sets give independence.
Yes
wdym independence?
@@peorakef with infinity it can be proved (through forcing) that A of C can't be derived.
@@ccdavis94303 so AoC isnt trivial for finite sets, it is not needed in the first place? only necessary for infinite sets?
@@peorakef yes
trivial in the sense that you can derive it.
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.
Yes exactly. The axiom of choice is about all collections of sets. This is made abundantly clear in the video, your ranting about finite families are not relevant to either the axiom of choice or this video.
sorry for necrocommenting, but I still don't get it. If we can prove that it holds true for finite sets, what's the problem to take some finite subset of infinite set? I mean then we can prove this statements by two lemmas: if we can pick element from non-empty subset then we can pick it from initial set, since initial set contains the subset that contains "pickable" elements. Then by subset axiom we can pick subset from initial set and it is done? I don't claim this to be right since I'm not good at set theory, but still I wanted to know if there are some problems with this proof
@@learpcss9569 I think you are confusing the size of the family with the size of the individual members of the family. When the family is finite, for example imagine a family comprised of 3 nonempty sets (say, A, B, and C), then it is possible to prove choice without using the axiom of choice, you can simply do induction. However, you cannot do this if the family is infinite (A, B, C, D, ...), you need an extra assumption such as the axiom of choice. Notice that we did not specify the sizes of the individual members A, B, etc., it does not matter if they are finite or infinite, they just have to be nonempty.
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.
@Trevor Chase AOD seems to involve more complicated concepts in it's statement than AOC. I guess I see this as better reason to believe AOC. However, I am saying this as someone who has never heard of AOD before, and I do not know much about topological games.
@Trevor Chase He didn't claim it's a justification of AC, just a description.
The intuitive fact we know about finite, is that a finite "forall" cannot be generalized to infinite "forall".
@@santerisatama5409 Well, then we are agreeing.
If I remember correctly Gödel prooved that if ZF is coherent then ZFC is coherent too, ie there's no way to disproove this set of axioms if the first cannot be disproove. Then if one can find an incoherence in ZFC, ZF is incoherent. Maybe the C axiom just helped us to find such problems that wouldn't have been found without it. Though it's kinda the worst thing that could happen, because if ZF is false, most of our modern mathematics are to forget 😬
{a,b} is very different from (a,b). {a,b}={b,a} is unordered and (a,b)≠(b,a) in general, unless a=b.
If you're talking about what is shown at 4:00 you seem to have misunderstood. Here (a,b) doesn't refer to an ordered pair of a and b. It refers to the open interval between them, i.e. the set of real numbers x for which a
@@seneca983wasnt talking about that
@@sufronausea OK, what were you referring to then?
@@seneca983 0:28 functions are defined such that their members are ordered pairs, not unordered
@@sufronausea Ah, you were referring to that. That's true, though a definition using unordered pairs would still work in the case were the function is one set to another such that the sets are disjoint.
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.
Agreed. If my reasoning is correct, then one such ordering is to first sort by the fractional part, then the integer part.
To be more precise: define two functions I(r) and F(r) so that I(r) is the "integer part" and F(r) is the "fractional part":
* I(r) = floor(r)
* F(r) = r - floor(r)
For example, I(10.3) = 10, and F(10.3) = 0.3. For a negative-number example, I(-10.3) = -11 and F(-10.3) = 0.7. (Sorry if that seems counter intuitive, but I needed to have the property F(r+1) = F(r) in order to easily define "least element" later.)
Now we define a less-than operator
@@danmerget Your ordering is not a well-ordering since the subset (0, 1) of R has no minimal element under the ordering
@@bachkhoahuynh9110 That's why I specified that the interval had to be "non-empty". If we use your criterion (i.e. that any open interval (a,b) must be non-empty), then it's impossible for ANY set to have a well-ordering, which renders the phrase "well-ordered" completely meaningless.
PROOF: Suppose set S is well-ordered. Then given an interval (a,b) with a
@@danmerget I mean (0, 1) is a open interval in the standard topology of R, i.e., (0, 1) = {x in R | 0 < x < 1}. Well ordering requires every nonempty subset has the smallest element. For every a, b in (0, 1), we have a
@@bachkhoahuynh9110 But a major point in this discussion is that we're NOT necessarily using the standard topology of R; we're free to choose other orderings. And I'm not sure where you're getting "a
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
Do you really want to get into an argument about notation and whether 2-element sets can be considered tuples?
@@andrasfogarasi5014 two-element sets aren't necessarily equivalent to 2-tuples, since ordering doesn't matter for the former. {1,2} = {2,1} but (1,2) =/= (2,1) following convention. the wiki article on ordered pairs says the modern set-theoretic definition is from Kuratowski. (a,b) := {{a},{a,b}}, and also offers a few equivalent definitions. this lets us get around the degenerate case with a=b, where we find {a,b}={a,a}={a}. without the additional structure from ordered pairs (in general, n-tuples). in this case, ordered pairs are very different from 2-element sets!
@@andrasfogarasi5014 yes because order is not defined on arbitrary sets. and as such the function can fail to be a function since we can make it such that its not right-unique
Plus a function is a triple (A,B,g) where A is the domain, C is the codomain and g is the graph. He only showed us the graph. In his definition, an inclusion map would be the same as the identity...
@@pecfexfextus4437 It's an interesting anecdote but also it's understood implicitly by anyone who cares. As Andras said, it's a notation argument. It's implicitly understood to be an ordered set because otherwise it would make no sense. You might not choose to write it this way for a proof but for everyday conversation it is far clearer to assume the sets are ordered. He could have said that, but honestly how obtuse do you have to be to not be able to understand you can't swap the input/output of a function?
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.
🧦
I heard this example in a Frederic Schuller lecture iirc.
This well known analogy is due to Bertrand Russel. Yes, Schuller mentioned it.@@bsatyam
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.
These points reflect my thoughts
“Banach-Tarski” are the correct names
@@Smooth_Manifold Yep. Derped on the name, as well as a few typos.
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
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.
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.
I know, I apologise for oversimplifying. The margin of this video is too narrow )
@@MetaMaths thanks for the reply, it be like that sometimes
Not true. Sets are the atoms of human intuition. Categories are intuitive sets but are not sets in ZFC. The distinction is very important. Both categories and ZFC sets are two different facets of intuitive sets that serve different purposes.
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.
Espoecially since it clearly does _not_ coincide with the standard ordering.
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
That is correct
the quote at the end gave me the chills
lovely video mate
stay healthy
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.
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.
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.
You actually get all sequences of cardinal steps, or all finite sequences?
@@gernottiefenbrunner172 Yeah sorry. I am only talking about finite sequences.
What does it mean to "step them down"?
@@austingubbels I simply mean to add a down step to the end of all the sequences, but the down step cancels out with the up step at the end of every sequence.
The context is stepping on the surface of a sphere. Watch the Vsauce on the paradox if you want more details.
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...
Unfortunately whatever the set of axioms you choose, as long as it allows usual arithmetics, there are true statements that are unprovable. This is Gödel’s incompleteness theorem.
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)
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
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!
maybe u should become a constructivist :)
most numbers can not be constructed.
The least productive type of proofs are those that simply prove something without giving any other information, for example proving that something exists but without information regarding how to get it. The best type of proofs not only prove something, they also give additional information for using the proven statement to do calculations or additional proofs.
@@JonathanMandrake Indeed. The fixed point theorem is great, because its proof gives also a way to find the FP!
That's AoC for you. It assumes that you can choose things, even when there's no way to do so.
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?
I'm not specialized in logic or theory set, but I think a reasonable way to think of sets is as a kind of properties class.
In second order logic, you can quantify properties, and the information of any set "A" can be encoded in one single property "Ã" such that "Ã[x]" ⇔ "x∈A" (so it wouldn't be that crazy to formalize sets just as "A=Ã").
I'm a Math Degree student and I have a good foundation in logic, but it's a very extensive branch and I would recommend that you ask specialized people in logic and set theory. 👍
A Proposição "todo espaço vetorial possui uma base" é equivalente ao axioma da escolha 02:35
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).
What you defined isn't a well order. It is an order that has a least element. Well order means that _every subset_ has a least element. Consider for example (0, ∞) ⊆ [0, ∞). This does not have a least element. Thus ([0, ∞), ≤) is not well-ordered. Moreover it is impossible to define a well order on any continuum-size set. We can only prove its existence. This follows from the independence of the existence of a well order on ℝ from ZF (see Jech). Otherwise, if it was definable, we could prove its existence in ZF alone and that would be a contradiction.
@@imengaginginclown-to-clown9363 Minor correction, the independence result implies that ZF can't prove any particular formula is a well order, but it doesn't imply that every formula *isn't* a well order. Under the axiom of constructability (V=L), there is a specific formula which you can write down and then a specific proof which shows that this formula well orders the entire universe of sets. This formula still exists in ZF, it's just that the associated proof is invalid (since it assumes V=L, which isn't an axiom of ZF). Since Choice is independent of ZF, we can't prove the formula is a well order within ZF, but since V=L is independent of ZF, we also can't prove that it's *not* a well order.
Otherwise I agree with everything you said.
Do not treat non-computable things AS IF they were computable. Formal math claims things like large cardinals and unmeasurable infinite sets ARE computable in the limited sense that the algorithms of pure logic (e.g deduction, excluded middle), still work when given such things as inputs. But this is lunacy, and blatantly mistaken, because things like Banach-Tarski are EXPLICIT EXAMPLES of that very machinery breaking down and throwing errors when you do that.
The Axiom of Choice is guilty of this error. The "for every" in its formula goes infinite, and treats noncomputable things illegally "as if" they were computable. It's a flawed axiom.
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.
An open interval doesn't have a maximum, definitely not considering the usual order of the real numbers. What happens is that AC guarantees some other order for which the open interval has a maximum.
@@PefectPiePlace2 I think that the things you said are equivalent. I'm playing cards now. Hope to remember to explain. And to be right. Hahaha.
Of course that is false. The usual intuitive order is not the only way to order the real numbers.
Look at the natural numbers, for example, you can define an ordering by divisibility: a
@@PefectPiePlace2 wow. I really thought a lot, until I remembered the order guaranteed by the AC makes the set well-ordered, meaning every subset has a maximum or a minimum, you can adjust your order to be one or the other.
But I still think we could argue without that information, kind following the idea of the proof of AC ---> well-ordering theorem. If I remember it correctly, it uses some idea of colimit in the category of ordered sets or something like that.
@@PefectPiePlace2 about "they are certainly not equivalent, the latter is much stronger", the fact that the latter is much stronger definitely do NOT implies "certainly they are not equivalent". There are several results in Mathematics that shows two statements, one clearly stronger than the other, as equivalents.
Module theory has a bunch, especially because the properties about modules over a ring depend a lot of the properties of the ring as module over itself. But in all areas of Mathematics there is some results like that. It is one of the wonders of Math.
Good study!
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.
0:32 Just nitpicking here, but I think there should be a difference between ordered pair notation and set notation.
You are right, my bad. What do you think about the video anyway ?
@@MetaMaths I think it was a good video overall. I hadn’t understood what was so controversial about the axiom of choice until I watched it. It really doesn’t feel like the real numbers should have a well order to me, but I guess the axiom of choice essentially demands it.
@@MetaMaths I have a question which is not really related to this specific video. I came to this video cuz I had an argument with my friend over ZFC icarus set and absolute infinity, he said since ZFC icarus set resolves cantor's paradox along with many, ZFC icarus set is actually bigger than absolute infinity itself, but from what i read absolute infinite should be the epitome of all the numbers, it breaks the notion of size itself, but I can't find anything on the net for confirmation, so I decided to study into ZFC and came here, but I'm really new to infinite series and cardinality, so would you pls clear my doubt on which is bigger, ZFC icarus or absolute infinite? Also if you can give reasonings on how, that would be really helpful, also we have a debate and i can't loose cuz I'll have to give him a party if I do :')
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}}.
Thats still a set of sets but I agree with you in principle.
A function in lambda calculus is well defined, but is not a set of ordered pairs. And what you claim that an ordered pair _is_ is merely one way of encoding an ordered pair. Cheers!
1:34 Joke's on you. I paused the video to understand the notation, and it made sense to me.
Good to know we have mathematics speakers among us
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.
_"Accepting the axiom necessarily means there exists things you can't construct, which some may consider problematic."_
Yes, that's kind of problematic. Consider the powerset of the natural numbers. What is a subset of the latter? To define one, we need a rule, that is, a finite string of characters. Now the set of finite strings made of a finite set of characters is countable. The powerset of ℕ is said to be uncountable (Cantor's diagonal argument). That doesn't mean that the powerset is larger (it isn't) but that there is no rule which can tell us which finite string is a valid rule. So the powerset of ℕ can't be constructed - it has a kind of diffuse "existence".
For axiom of choice all the way 🎉
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
I meant that it contradicts our normal intuition
0:30 Note that tuple is also defined by sets. (a,b)={{a},{a,b}}. It can then be proven that (a,b)=(c,d) iff a=c, b=d by the definition of set equality
I have zero problems with the the BT paradox. But the "open sets have a least element" is the issue.
I can see the banach-tarsky as just equivalent to scaling, you are dealing with infinities after all.
But the coarseness of "least element of the open set" on a smooth set is the problem for me.
It gives a "least infinitesimal", which is a contradiction on itself.
My favourite thing equivalent to the Axiom of Choice is that every fully-connected graph has a spanning tree.
Why so complicated example? Every non-empty graph has a vertex.
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?
Obviously, the usual order on real numbers is not a well-ordering relation - for example, the open interval from 0 to 1 has no minimum under this relation. In fact, an exact opposite is true: the usual order on real numbers is a dense order - given any two different real numbers x and y there exists a real number strictly between them, such as (x+y)/2 (and by extension there exist infinitely many such real numbers).
Just... wow! 💜
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.
I have a genuine interrogation about category theory. From what I know (which is little), there does not exist of formalisation of category theory that is not based on set theory. Therefore I don't really see how there could be fewer problems, and I would be interested if you had a resource with for instance a list of axioms for category theory.
@@An-ht8so See Proceedings of the Conference on Categorical Algebra, La Jolla 1965. Professor Lawvere has published a paper there called "The Category of Categories as a Foundation for Mathematics" which seems to be what you are asking for.
No, you are wrong, set theory is able to express all of the current mathematics. You can indeed formalise first order logic inside of set theory and this is indeed done in practise when working in logic. The question of "what came first" is irrelevant to the question of "can it be done in set theory".
The thing about classes is also a bit redundant, we usually only care about definable classes and can handle them similarly to sets, because we can say x ∈ C iff φ(x) with a slight abuse of notation and work as usual. If you really need a set theory with classes you can also use NGB.
"Fewer problems" is also a bit vague. Type theory sure is great for formalising with a proof assistant. Otherwise, we can interpret most type theories in ZFC and ZFC within most type theories, so that's not really "better" or "has less problems". I also don't see how you can claim that category theory has less problems as a foundation when this is not really a developed theory.
Where does the cover from the video comes from ???
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 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 TH-cam comments about freewill’s irrelevance, pat yourself on the back and leave it at that.
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 '
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?
Our typical way of ordering the reals is not well ordered, but that is not to say that there is an ordering that works completely differently that is in fact well ordered. The same is true for the rationals. By taking our ordering to be the standard then the rationals are not well ordered. But we can put a different ordering on the rationals so that they are well ordered because they are bijective with the natural numbers.
@@robertgamer3112 Interesting. I am currently taking a proofs class, and we just did the diagonalization proof that the reals are uncountable. How does that proof depend upon the size of the numbers? I understand what you are saying about the rationals, but I am still a bit confused
@@Jesse_Carl What he is saying, is that for a countable set such as the rational, you do not need a powerful theorem to find a well ordering, as you can transpose the well ordering from N to your set. For the real numbers you cannot do that, and the axiom of choice is needed. However the proof does not depdend on the size of the set in the sense that you could use the axiom of choice to find a well ordering of Q, although it would be needlessly complicated.
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.
> 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"?
Sure, but we do that because every once in a while you'll get a perfectly crafted cheesecake out of it, which makes absolutely no sense and yet is somehow perfectly consistent. Its those completely unexpected results are what moves the highest ends of math forward.
"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"?"
Nonsense. This is only "too much of pure math" if your opinion of pure math comes from popsci sources talking about the foundational problems people were grappling with in the early 1900s.
"and the fact you still get paradoxes is really a reflection of the fact that true reality doesn't allow infinite things."
This is good insight. All "paradoxes" resulting from the axiom of choice depend on the axiom of infinity as well. And as some like to point out, sometimes rejecting the axiom of choice has results that are equally "paradoxical" to accepting it. For example, it is consistent with the negation of the axiom of choice that the set of real numbers can be partitioned into strictly more nonempty subsets than there are real numbers.
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?
the existance of the empty set follows from the axiom ∃ x (x=x) and the comprehension schema. Essentially once anything exists, so does the emptyset
What’s the background music?
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
Love your video🤓🤓
Please do a video on topology
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.
Amazing. Do you have any books to recommend?
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?
Problems? Yes. The order that you defined is not a well-order, because not every subset has a least element.
@@YAWTon Aah, Now I see where is the tricky part, thanks :)
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.
Why do you see this as an error ? The idea of this video is to present this decision as a philosophical challenge.
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?
Yes. If the input to the choice function (implied to exist by AC) is the set of all non-computable irrational numbers, then the output (i.e. the choice) will be a member of that set, i.e. a non-computable irrational number.
Doesn’t non-computability imply transcendence?
@@headlibrarian1996 Yes. I was just quoting from the original post. "Irrational" is (very) redundant in the phrase "non-computable irrational number" ; )
@@normm5025 How do you build the set of all non-computable irrational numbers?
@@andsalomoni I don't know what you mean by "build". You can arrive at them by removing the set of computable numbers from the set of all reals. Would you say you could build the set of all computable numbers? If not, then you also can't build the set of all non-computible numbers.
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
But AD has itself weird consequences for infinite combinatorics. However "Definable Determinacy" is "true" in the sense that it follows from large cardinals and it is consistent with choice. This is enough to get reasonable behavior for reasonably defined sets. It excludes projective Banach-Tarski decompositions for example.
There is a mistake in the first order statement, it should say [... Union of A forall A in X (f(A) in A)]
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 ❤🍀👍
Set theory is precisely what we call a theory in logic. There can be other theories than set theory. There are theories that are stronger than set theory, in which we can prove more things. For you last question, one important theorem of logic is one that states that if a theory is consistant, i.e. has no contradiction, then this theory has a model, i.e. there exists a mathematical universe whose objects obey the axioms of the theory.
Set theory can and does treat mathematical logic. For example the compactness theorem mentioned in this video* is a theorem of mathematical logic.
It depends on what you want to choose as your foundation if you want some "more fundamental" structure than a set. There is for example type theory as well.
Classically foundation is viewed and structured like this:
1. Establish the language of first order logic
2. Define inference rules (for example sequent calculus or natural deduction)
3. Formulate ZFC in this language
4. You can do all the maths in ZFC from there on
*the equivalence shown in the video is an open question, why it is presented as a fact is beyond me
Excelent video! Could you tell me what progam do you use to edit videos?
The graphics appear to be done with manim, and the video editing itself is quite basic, it could be done in any editor.
@@tissuepaper9962 Thanks man.
You're stating that any mathematical thing can now be described as a set, however, there are some sets that actually cannot exist even though they are conceptually like sets and if this is the case, how could all of math be in set theory if set theory itself cannot fully handle set theory itself? For example, say I want to make a set of all possible sets. Before that, I examine the opposite: One such set would be a set that has all sets or elements that cannot be put into a set. But, this set is self-contradictory. So, if we do not allow for contradictions (which may be allowed but then leads to a lot of issues) Then a set that has all sets or elements that cannot be put into a set leads to self-contraction in infinite cases. But we said earlier that there is a set of all possible sets. So, now we must either admit that assuming anything can be a set leads to contradictions or admit that it is fine to create a set of all possible sets but that there are some thing that just cannot be sets. Now, if we extend this logic by replacing the words sets or elements with branches or theories of math, then can see how it is not possible for set theory to contain or describe all of math.
your youtube chanel is just incredible, perfectly balanced !
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}} }
{a, b} is not the same as (a, b) 0:31
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.
If the axiom of choice seems obviously true to you, consider:
With suitable definitions of what (finite number of) symbols and (finite number of ) operations we can use, we can sensibly define a property on real numbers "has a description of finite length". I.e. the numbers for which there is a way for me to tell you exactly what the number is, which will not take forever.
Now consider the set "real numbers which do not have a description of finite length." Can you choose an element out of that set? If so, which one did you choose? (The set of numbers with a finite length description is countable, reals are uncountable, so this set is not empty.)
Is choice needed for indescribable numbers to exist?
@@MuffinsAPlenty No, indescribable numbers exist because describable numbers are countable and reals are uncountable. This is just an example of a set where the Axiom of Choice is intuitively false, rather than intuitively true.
@@michaelwoodhams7866 Then this is not a good example against the axiom of choice. You don't need the axiom of choice when you're choosing a single element from a single nonempty set. That's provable from the non-choice axioms of set theory.
This is, perhaps, a nice argument in favor of some form of mathematical constructivism (which often does reject the axiom of choice, but also often rejects other logical laws too).
@@michaelwoodhams7866that is not correct. Describable numbers are countable in an intuitive sense, whereas reals are uncountable in the set theorical sense. Countable in set theory is not the same as intuitively countable. Because non standard arithmetic exists, N does not have to be countable in the intuitive sense.
In fact, the skolem theorem guarantess that there exist models of set theory with countably many set (intuitively), it's called the skolem paradox.
The "set" that you are talking about does not really have a mathematical definition, so you can't really talk about it at all.
Beside, and most importantly, if you could define this non empty set, and name it E, I could just write "Let be x in E" to pick an element from E, without the need for the axiom of choice.
@@MuffinsAPlenty OK, thanks. I'm an ex-astronomer who sometimes pretends to be an applied mathematician, so I'm clearly out of my depth here.
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
Wait, what? Is the well-ordering theorem equivalent to the axiom of choice? In that case both must be false?
Well ordering implies axiom of choice
@@MetaMaths ah ok. Well a false proposition implies anything, doesn't it? Ex falso quodlibet.
@@kevalan1042 OK, but I am not sure what you mean by axiom of choice or the well-ordering theorem being false. Axiom of choice is equivalent to the proposition that all sets can be well-ordered. And set theory without axiom of choice can't prove axiom of choice true or false (that is, assuming set theory itself is consistent). There's a model of set theory (such as the constructible universe) where axiom of choice is true; and there's a model where axiom of choice is false.
Well-order theorem isn't trivially false. In fact, AC => WO, and WO => AC, which means it is independent from ZF.
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}
I agree, the burden of this error will pursuit me forever... Thanks for being attentive ! I love careful viewers.
I always assumed the term 'tuple' to imply ordering.
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
Is the category of small categories a set?
I'm pretty sure it can't be just like the class of all sets isn't a set.
I never understood that final quote. All three seem equally obviously true to me.
I'm team axiom of choice all the way ! just cuz somethings weird to us doesn't mean it's wrong
More like obvious but not redundant as it may seem
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.
The Banach Tarski paradox specifically talks about isometric transformations, i.e. taking the pieces and moving them about or turning them without deforming, shrinking or expanding them in the intuitive sense. That is: only acting on them by orthogonal matrices and translations.
Also: "Zorn's lemma is obviously true" LOL
@@lukasrollier1004 Does it happen in a space we can't imagine then, similar to that hypersphere that's boxed in by touching hyperspheres in the corners of a hypercube, and is actually larger than the corner hyperspheres with enough dimensions? Cause what you describe sounds like doing it in 3d with finitely many parts should be blocked by some invariants unless i'm misunderstanding something
In a dimension higher than three, we can take isometrics that only turn in the dimensions we're used to and leave everything else intact, so higher dimensions follow from the regular Banach-Tarski paradox.
Also "Zorn's Lemma is obviously true" LOL
@@lukasrollier1004 That doesn't mean that the product of two of those orthogonal matrices have to be a 3D turn, or that the same invariants exist
No, the (tensor)product or two such transformations would be 9-dimensional. I only suppose that we turn in three dimensions and leave the rest untouched.
Btw: what was that about Zorn's Lemma?
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.
can you do a video on first order logic?
We need the Axiom of Countable Choice to prove for continuity the equivalence of the epsilon-delta definition and converging series definition. That axiom does not lead to paradoxes like Banach-Tarski. There's also the Axiom of Dependent Choice, as slightly stronger formulation than countable choice, but still no paradoxes.
@Trevor Chase - have you worked with the Axiom of Dependent Choice? I only took the normal calculus course of year 1 and 2. We used the Axiom of Choice occasionally, which was always specifically announced, as it should, I think. But we never mentioned Dependent Choice.
Well I just feel it is important to point out that the Banach-Tarski "paradox" is not actually a paradox - it is just a very nonintuitive result that "feels wrong". Also I believe if you limit yourself to the axiom of countable choice you might lose the ability to work with things like function spaces (?)
@samuelallanviolin752 - many different sets of axioms are possible starting points to do mathematics. Using the unlimited Aciom of Choice does not lead to internal contradictions, that's true - but if it leads to the Banach Tarski paradox or the "obviously false" well-ordering theorem, how good is it as starting point? Your conclusions from it will remain odious. That's no problem if you stay within mathematics, but what about convincing non-mathematicians of your usefulness and your sanity?
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
I suggest that all mathematics presupposes some pretheoretical notions -- a collection of things being one such notion. Ie sets, but in a naive rather than 'set-theoretical' sense. So sets are involved, but not set theory.
Set theory will probably not help nor hurt you in getting wherever you need to be. Algebraists, topologists, number theorists, what-have-you-ists, definitely don't deal with these foundational issues (They usually just pick whatever choice axiom that makes their job the easiest. For example if you're working in Linear Algebra you would accept AC since it gives that any vector space has a basis.) but it's not the worst thing to do with your time.
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.
As far as I can see, this should work, for open and closed intervals. But I'm not a logician.
The problem with well-ordering is that ANY set of real numbers should have a well-ordering.
You need a minimum for _every_ nonempty subset, so that there exists no strictly decreasing infinte sequence of elements at all. So you need to start numbering all the elements of the set, the first one, the second one, ..., the omega-th one etc. We need to count far beyound the natural numbers (transfinite recursion) and in a sense choose arbitrarily which element we want to assign the next ordinal number. There is no definable wellordering of the reals.
If you can well-order a set X of size continuum (for example open or closed non-empty intervals) then you can well-order ℝ. There is a bijection f: X → ℝ. If we call the well-order on X by the letter R then we can define a well order S on ℝ by f(x)Sf(y) iff xRy for all x,y ∈ X. It is well known that ZF (meaning no axiom of choice) cannot prove that there is a well order of the reals (see for example Jech); so you cannot define one, because otherwise it would provably exist in ZF.
@@imengaginginclown-to-clown9363 I came to this video cuz I had an argument with my friend over ZFC icarus set and absolute infinity, he said since ZFC icarus set resolves cantor's paradox along with many, ZFC icarus set is actually bigger than absolute infinity itself, but from what i read absolute infinite should be the epitome of all the numbers, it breaks the notion of size itself, but I can't find anything on the net for confirmation, so I decided to study into ZFC and came here, but I'm really new to infinite series and cardinality, so would you pls clear my doubt on which is bigger, ZFC icarus or absolute infinite? Also if you can give reasonings on how, that would be really helpful, so do you know something on this topic? Would really help me out :)
@@gauravdoley1527 I'm not sure what sort of fringe pseudo-math your friend has been reading but he is certainly on the one side either wrong or not even wrong (in the sense that what he says does not make sense).
First of all the not even wrong part: Absolute infinity is a concept from philosophy and it is not defined _at all_ in ZFC. If something like this _would hypothetically_ exist then it would be some sort of super Ord and an Icarus set would not be larger than that. But it is not even defined so the question is entirely ill-posed.
I'm not sure what Cantor's paradox refers to here; it usually refers to the fact that some classes of sets are not sets, like the class of all sets (there is no set X such that for all sets Y, Y ∈ X). It isn't really a paradox in the sense that there is a contradiction here. If the existence of an Icarus set would imply that this is not the case then Icarus sets are inconsistent with set theory. This however is currently unknown since consistency of Icarus set is equivalent to consistency of I0.
If you want to learn some set theory you should find a proper book about it. Consider for example Jech's set theory.
anyone ever noticed that the AoC is beraking the rules that logics self-imposed onto it?
as Wttgenstein noticed, it is a silly game.That does not mean that it could not be corrected, yet the ZF formulation is, as said, silly.
I'm not sure why people give a negative connotation to paradoxes. The "Reals" of set theory are not the Reals because they're just a set, without an algebraic structure, or any ordering. People might as well be surprised that there is a well ordering on the powerset of the integers. Paradoxes should remind people that they haven't understood correctly what they're talking about, that very likely superimposing some structure that isn't there
- 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.
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?
The doubt comes from our intuition, if you like. Certain consequences of AoC that are mentioned in this video appear counter- intuitive
_"Could someone please explain why/how there is any doubt over the Axiom of Choice being true?"_
It gets kind of diffuse if infinitely many sets are involved. Let's say that we partition the interval [0, 1] in several sets of points. We can do that by defining that any two points should belong to the same set if their distance is a rational number. Quite easy a partition, isn't it?
But now we create another set which consists of exactly one point of each of these previous sets. How do we choose this one point? We need an algorithm to do that. Unfortunately, _there is no such algorithm._
Now that's where things get ugly. Some math people say: we don't need no fckg algorithm, because the axiom of choice tells us that this set does exist: we _postulate_ that the set exists. - But other math people say: no algorithm, no set. Peculiarly, they are the minority.
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.
3:16 if two theorems are equivalent, how can one be weaker than the other?
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.
You only need look where the answer are. We first need to critique with the historical hammer.
It is not true that every mathematical object is a set.
I am not evern sure that every mathematical object can be modelled as a set.
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?
I think without the axiom of choice, you are forced to work with boring vector spaces in maths like R^n where you are able to imagine a basis.
Just imagine the vector space of all bounded functions from fixed set M to R, so {f: M->R | there exists C>0 with f(M)
After seeing that every things we define is based on assumptions called axioms, I am having a mathematics existential crisis.
You are in good company. Check how Hilbert felt about Gödel's incompleteness theorems.
Axioms of ZF are merely definitions of what a set is. Every definition is formaly an axiom. If you're not puzzled by definitions, you should not be puzzled by axioms.
@@michaeldamolsen Hilbert knew about the thing that Ranjit mentions here. The incompleteness theorems go a lot further than this.
@@An-ht8so That's not true. Definitions are usually shorthands for longer formulas while axioms limit the class of models which satisfy your theory.
@@imengaginginclown-to-clown9363 What you're describing is the difference between conservative extension and proper extension of a theory, but they're still langage/theory extension, new symbols, new axioms. They can be shorthands, but they can also have real theoretical value. For instance, Henkin witnesses are absolutely core to proving the completeness theorem, The truth is "definitions" are not a logical concept
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.
You can prove that a statement is unprovable (goldiel did this with his incompleteness theorum i think). I think that's what metamaths meant
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.
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
You don't need to be a believer, you can be a formalist: Assuming the Axiom of Choice for this class, how far can we get? For another class: Assuming the Axiom of Dependent Choice, how far can we get now?
@Me Too - what's wrong with a game, especially such a beautiful and useful one as mathematics? Or do you think seeing math that way would disrespect its truth?
. . . In history, formalism has probably been the leading choice of mathematicians since 1900, following Hilbert and Bourbaki.
@Me Too Math is just a chess game. Axioms are treated no more than the fact that chess pawns can only move one square at a time-a fact that you don't believe nor refuse, but accept simply because the rules said so! :)
Sounds officious to me. I would stand up and say "I do not. Does that preclude me from getting an A?"
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.
Why can't it just be added to the axioms?
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.
Cantor diagonalization proves that the set of reals is uncountable, i.e. not in bijection to the natural numbers. Look up ordinals to find a way to number past infinity.
The video is misleading in its explanation of the well-ordering principle; it should be more clear about the fact, that the principle orders the reals in a very different way than the usual ordering; simply pick a number after the other ad so much infinitum that you cannot imagine and then some more. Again look up ordinals to get a sense of this.
The following statement makes no sense: "it is equivalent to a slightly weaker assumption". If the two statements are equivalent non of them can be weaker than the other. A weaker statement can only be implied by a stronger statement.
It's a weaker-looking statement.
"...which side are you on?"
Do I have a choice ?
The concept on infinity leads to very strange paradoxes. If you want all of mathematics to make sense, you should keep within the safe realm of finite sets and stay away from the AoC.
Another problem arises from the axiom of the “excluded middle” (all statements are true or false).
I really think you should do a video on “constructionism” or “intuitionism”, which is the mathematics without the excluded middle. Only direct proofs are allowed (A -> B -> C etc.), or put differently, to prove C, it’s not enough to prove that not C leads to a contradiction. You have to prove that C follows from the axioms.
Every existence proof therefore also leads to a construction of the element.
What is a vector space?
We could use Axiom of determinism to solve all problems axioms of choice claim to be solving and not fall into paradoxes.
Sorry, I am too busy splitting the continuum into more than continuum-many disjoint, non-empty subsets! Or, as I like to visualize this: imagine Cantor's Hotel - competitor of Hilbert's Hotel, where rooms are indexed by real numbers. The hotel has infinitely many floors, and infinitely many rooms on each floors: on each floor there are numbers which differ from each other by a rational number. In absence of axiom of choice there can be more floors than rooms! ("Say what! Of course there are no more floors than rooms." Okay, what does this mean? "It means that floors can be mapped one-to-one with a subset of rooms." And can you do that? "Sure, just pick a single room from each floor..." Oops. That's precisely what axiom of choice says. And, in fact, axiom of determinacy makes this impossible!)