The Axiom of Choice
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- Опубликовано: 25 май 2024
- Mathematics is based on a foundation of axioms, or assumptions. One of the most important and widely-used set of axioms is called Zermelo-Fraenkel set theory with the Axiom of Choice, or ZFC. These axioms define what a set is, which are fundamental objects in mathematics. And the Axiom of Choice is arguably one of the most important and interesting axioms of ZFC. But what does it really say? And how is it used? This video dives deep into the formal definition of the Axiom of Choice, as well as its important equivalences which have their own fascinating applications in various branches of mathematics. Furthermore, we look into the controversy behind AC, and why it has garnered much discussion throughout its mathematical history.
0:00 Introduction
1:28 Set Theory and ZFC
9:22 The Axiom of Choice
16:55 Zorn's Lemma
23:39 The Well-ordering Theorem
27:49 Other Equivalences of AC
29:23 Controversy & Final Thoughts
Additional Resources:
The Banach-Tarski Paradox by Vsauce: • The Banach-Tarski Paradox
Wikipedia article on the Axiom of Choice: en.wikipedia.org/wiki/Axiom_o...
Wikipedia article on ZFC: en.wikipedia.org/wiki/Zermelo...
Music:
c418.bandcamp.com/album/dief
Smooth Fall by C418
Work Life Imbalance by C418
c418.bandcamp.com/album/life-...
In Berlin people act differently by C418
c418.bandcamp.com/album/seven...
The first unfinished song for the Minecraft documentary by C418
patriciataxxon.bandcamp.com/a...
Cribwhistling by Patricia Taxxon
Starboard by Patricia Taxxon
Animations were made by Manim, an open-source python-based animation program by 3Blue1Brown.
github.com/3b1b/manim
You do not want to know how long I've wracked my brain for how to define ordered sets in set theory. Thanks for putting that in this video and releasing me from my stupidity in not looking it up properly.
I would consider set theory math as last generation, math in category theory/type theory is the future😅
@@kaa1el960that's just what the category theory propaganda wants us to think!!
@@Gabriel-nw6fc haha that's true, but once you go there you'll never come back. Luckily I began learning all math in using sets, so I can switch back and forth.😂
This is duality;
Injective is dual to surjective synthesizes bijection or isomorphism.
Absolute truth is dual to relative truth -- Hume's fork.
Syntax is dual to semantics -- languages or communication.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory.
If mathematics is a language then it is dual.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are dual!
The integers are self dual as they are their own conjugates.
"Always two there are" -- Yoda.
Subgroups are dual to subfields -- the Galois correspondence.
--wracked-- racked
love the c418 music, it goes way better with math videos than I would've thought
One thing that is quite important but often skimmed over is that axioms are not meant to be "obviously true". They are true within the framework of a theory, they are the "ifs" to the "thens" the theory shows. A theory does not assert them, it *requires* them. We needed to abandon this particular mindset that there is a single universal "mathematical truth" to discover so many new worlds, from complex numbers or infinite cardinalities to non-Euclidean geometry. If you find a situation where the axioms are actually true (like the real world), then you can happily apply the theory, but if not, it is equally valid to use it in a "what if" way.
I appreciate that you gave a basic understanding of the terminologies, which where to be used in a statement. And as a person who forgets everything, I really appreciate your recap. Thanks.
Hello from France ! Please keep create, your videos got sooo much potential (and they're really interesting)
The first time I had seen the Cartesian product, I was like “what is the point of this” and then I stumbled upon the innocuous looking statement that “a relation is a subset of a Cartesian product” and found out that it was the Way itself, the vessel that is inexhaustible because it is empty.
Huh yeah, I guess a function is just a set of pairs of items from the domain to the range, which is a subset of D × R
@@redpepper74 yep, a relation from A to B is a subset of A × B, a "total" relation has at least one b ∈ B related to any a ∈ A, a "univalent" relation has at most one b ∈ B related to any a ∈ A, and a function is a total univalent relation (exactly one b ∈ B for each a ∈ A). A partial function is just a univalent relation.
@@samsamson3315 would "total" be a similar concept to surjectivity?
ok Lao Tsu
This is duality;
Injective is dual to surjective synthesizes bijection or isomorphism.
Absolute truth is dual to relative truth -- Hume's fork.
Syntax is dual to semantics -- languages or communication.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory.
If mathematics is a language then it is dual.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are dual!
The integers are self dual as they are their own conjugates.
"Always two there are" -- Yoda.
Subgroups are dual to subfields -- the Galois correspondence.
Holy mackerel, this was an amazing video! You covered _a lot_ of ground in only 32 minutes, with very clear exposition. I'm most impressed.
This is duality;
Injective is dual to surjective synthesizes bijection or isomorphism.
Absolute truth is dual to relative truth -- Hume's fork.
Syntax is dual to semantics -- languages or communication.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory.
If mathematics is a language then it is dual.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are dual!
The integers are self dual as they are their own conjugates.
"Always two there are" -- Yoda.
Subgroups are dual to subfields -- the Galois correspondence.
These are great videos, it’s only a matter of time until you blow up imo, keep up the good work
your consistency in producing quality content is admirable!
Nicely done. Just the right level of detail.
Something is wrong. How come I wasn't subscribed to this channel yet!!?? This video is great!!!
really nice to hear Patricia's music in here, i was surprised!
and lots of C418, love it
@@weirdredstone42 love the c418!!
Great video man, you are gonna blow up for sure!
I really enjoyed your presentation style. Very informative and coherent. Thanks!
great video on a very interesting and profound subject! thanks!
Great work as always, keep on doing what you do👏
This is what I call Maths.
We need more content like this.
This is duality;
Injective is dual to surjective synthesizes bijection or isomorphism.
Absolute truth is dual to relative truth -- Hume's fork.
Syntax is dual to semantics -- languages or communication.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory.
If mathematics is a language then it is dual.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are dual!
The integers are self dual as they are their own conjugates.
"Always two there are" -- Yoda.
Subgroups are dual to subfields -- the Galois correspondence.
incredible work
First time wacthing your videos and already subscribed!
Great explanation and very clear. Thx.
This is duality;
Injective is dual to surjective synthesizes bijection or isomorphism.
Absolute truth is dual to relative truth -- Hume's fork.
Syntax is dual to semantics -- languages or communication.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory.
If mathematics is a language then it is dual.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are dual!
The integers are self dual as they are their own conjugates.
"Always two there are" -- Yoda.
Subgroups are dual to subfields -- the Galois correspondence.
To understand why the axiom of choice is controversial, consider the hat-puzzle of the last decade. An infinite list of prisoners have a hat placed on their head. Each prisoner can see all the others, and has to guess the color on their own head. The prisoners "win" if all but finitely many guess correctly. The axiom of choice on the reals allows the prisoners to "win". This is incompatible with the idea that each hat choice is indepedendent. So the axiom of choice is restricting the form of power-sets to be in some sense "correlated" or "determined", and this is made precise by defining the Godel model "L", where each infinite subset is determined by a rule from a pre-existing ordinal.
Is there a way of resolving this?
@@Happy_Abe It depends on how much logic you are willing to learn. The "cheap and easy" fix is to assert that every subset of R is Lebesgue measurable, this is implied by things like the axiom of determinacy, so you can go ahead and assume that. That's good enough to restore intuitions for R, but then you get counterintuitive propositions about maps into R at higher levels of the set theoretic hierarchy. Those don't matter for day-to-day mathematics, but they do matter if you want to fix the foundations properly.
@@Happy_Abe The limit where set theoretic forcing makes statements alterable is at what is called the Schoenfeld absoluteness limit. This is at what are caled delta-1-2 statements, statements with two real-number (or function) quantifiers in second order arithmetic. Beyond this, there is no hope of making a system where you can't choose truth values for certain sentences, although certain large cardinals allow you to arguably push the limit higher.
@@annaclarafenyo8185 this is all very interesting. Yeah I was interested in fixing the foundations properly. Of course none of this matters for most things you’ll be concerned with but foundationally these things are important.
I’m actually considering taking a logic class next semester in my grad school but some are suggesting I shouldn’t unless I really want to go into logic. There’s a forcing in set theory course, lots to choose from lol
@@Happy_Abe You MUST take logic, but it's insanely "expensive" in terms of concentration required. The trick is to understand that it is talking about computations, that computations are fundamental. To learn proper foundations, start with a "HoTT" book or course, to learn how to program proofs (this is a quick and dirty way of learning both first-order-logic and also intuitionistic logic, which is the 'correct' logic, but it's really hard to understand how to do set theoretic stuff in it). The main highlights of logic are: 1. Computation and logical completeness, this is the 'world's worst programming language', consisting of "or", "and", "implies" and bounded arithmetic predicates. It's terrible, but you have to learn it. It's equivalent to programming in C with infinite integer variables but only bounded "for" loops which are not allowed to change the loop variable. General computation just adds the ability to loop forever. 2. Godel incompleteness, learn how to prove it computationally. 3. Turing oracles: this is essential, to translate to logic transitions to second-order arithmetic (you need to be able to talk about reals) 4. Hypercomputation: this is what happens when you iterate ordinals over Kleene's O, it's equivalent to Delta-1-1 predicates in second order arithmetic, or to Turing computation with the added magic that you can always tell when the computation is going to halt or not.
The proper foundations is going to be something analogous to HoTT, but using Spector's 1962 posthumous paper on the computational formulation of second order arithmetic using "bar recursion" (well-founded recursion, or 'arithmetic transfinite recursion' in second-order arithmetic jargon). Set theories can be coded up in this language, but questions like the axiom of choice become irrelevant for collections as big as the reals, because they aren't present in second-order formulations.
The 'axiom of choice' is just true when it's dependent choice, the thing about uncountable structures is that they are not absolute, they are a dogma, you can modify their properties nearly entirely arbitrarily using forcing. The standard expositions of forcing are not as good as Nik Weaver's "Forcing for Mathematicians". Read that. And Cohen's original book. BUT, you need to learn forcing in the context of computation and second order arithmetic too, this was pioneered by Steele in the 1970s with "Forcing on Tagged Trees", and perhaps there is a more recent reference elsewhere.
You make truly superb content keep it up.
fun topic and cool animations, thanks!
Amazing video.
None of the equivalencies mentioned here requires the axiom of replacement, neither Zorn's lemma nor the wellordering principle. ZC is enough. While replacement allows a mapping to ordinals rendering the induction more intuitive, we can technically do without them. Historically Zermelo introduced AC to prove the wellordering principle decades before Fraenkel added Replacement: ZFC. The Bourbaki group based the early versions of their books on (something like) ZC.
Most of basic, classical mathematics does not require replacement. Only when the existence of larger sets is needed (Borel Determinacy being a prime example) replacement is required. ZFC proves the consistency of ZC, so it is a strictly stronger theory.
The equivalences are provable over Z. Proving them over ZC is overkill, since the theorems themselves are already provable there.
You're right, I was more so talking about that specific proof of these equivalences rather than the actual equivalences themselves. For example, there are proofs that don't even use ordinals or transfinite induction (www.jstor.org/stable/2323807?seq=1). But since the most common proofs used in undergraduate curriculum (as far as I know) use transfinite induction and axiom of replacement, I thought they were worth a mention.
This is duality;
Injective is dual to surjective synthesizes bijection or isomorphism.
Absolute truth is dual to relative truth -- Hume's fork.
Syntax is dual to semantics -- languages or communication.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory.
If mathematics is a language then it is dual.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are dual!
The integers are self dual as they are their own conjugates.
"Always two there are" -- Yoda.
Subgroups are dual to subfields -- the Galois correspondence.
@@hyperduality2838You ok?
From a technical viewpoint, the axiom of choice matters because things in math can only be defined in a finite number of steps, it can deal with infinite objects but only in a finite way. That's why the existence of the set of natural numbers, an infinite set, must be a new axiom, unable to be proven from previous axioms.
This is duality;
Injective is dual to surjective synthesizes bijection or isomorphism.
Absolute truth is dual to relative truth -- Hume's fork.
Syntax is dual to semantics -- languages or communication.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory.
If mathematics is a language then it is dual.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are dual!
The integers are self dual as they are their own conjugates.
"Always two there are" -- Yoda.
Subgroups are dual to subfields -- the Galois correspondence.
this was cool and inspiring
One of the best out there!
Two key points in the video were when you mentioned ordinals and chain
Earned yourself a subscriber. Amazing video!
I’m taking real analysis, algebra, and algebraic topology now in grad school and we use Zorn’s Lemma all the time. Thanks for making this video and helping me understand this concept better!
This is duality;
Injective is dual to surjective synthesizes bijection or isomorphism.
Absolute truth is dual to relative truth -- Hume's fork.
Syntax is dual to semantics -- languages or communication.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory.
If mathematics is a language then it is dual.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are dual!
The integers are self dual as they are their own conjugates.
"Always two there are" -- Yoda.
Subgroups are dual to subfields -- the Galois correspondence.
30:15 Well, you don't need much for Banach Tarski actually. Terrence Tao showed how to use hyperreals which need only ZF + Ultrafilter Lemma, which is strictly less powerful than Choice, in order to construct unmeasurable sets.
Actually, you can prove Banach Tarski using only the Hahn Banach theorem which is even weaker!
I wish to be this smart one day. :)
I would like to know in which branch if mathematics does this come under?
I am currently in freshman year of University, and in next semester I'll have discrete mathematics which also excites me!
@@Redditard omg it's my current thoughts right now, also 1 semester at uni (actually the 2nd as of 1st of April)
@@Redditard Hahn-Banach and non-measurable sets are usually wrapped up with real analysis. For talk about Ultrafilter Lemma and all that you need to go into logic. Banach-Tarski is… more of a curiosity than anything, so it doesn’t really fit in a course. The VSauce video about it is very good, though.
@@TheBasikShow hmm, I'll look into it... Thanks!!
@@user-nm7gb3rw9c Lol, it's my 2nd semester rn and third will start in June... We will have Discrete mathematics in our syllabus! And I am so excited for itttt
bro your videos are bussin
On god frfr
Great video, thank you for your work!! New sub here. Greetings from Argentina
Great Work! ^.^
I still cannot see why the BT paradox is not a proof by contradiction of the AC. What an impressive video, clear concise. How can it be possible to well order the real numbers, when each has infinitely many digits.
This is duality;
Injective is dual to surjective synthesizes bijection or isomorphism.
Absolute truth is dual to relative truth -- Hume's fork.
Syntax is dual to semantics -- languages or communication.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory.
If mathematics is a language then it is dual.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are dual!
The integers are self dual as they are their own conjugates.
"Always two there are" -- Yoda.
Subgroups are dual to subfields -- the Galois correspondence.
@@hyperduality2838 are you an AI
What does it contradict? If you can't explicitly derive a contradiction, there's no proof by contradiction.
I don't find BT any more troubling than Hilbert's Hotel. Like, yeah, that's the kind of thing you can get with infinite sets. You can break them into pieces that each can be mapped back onto the original.
Banach Tarski isn't equivalent to Choice, it's strictly weaker. It's implied by Hahn Banach, and you _can't_ get rid of Hahn Banach. Ain't happening.
Great video. You are doing a fantastic job explaining these concepts. Thank you. May I suggest you use a music with less dissonance and less use of high frequencies? It can get distracting.
It is just a suggestion, you're awesome dude. Keep up the good work!
love this
what book/lectures do you suggest to go into more details about ZFC set theory ?
great video, new sub added!
Good video, but several typos throughout, especially regarding names (for example 2:30 has Russel instead of Russell, 4:20 has Bernsays and Godel instead of Bernays and Gödel). You also pronounced "Neumann" as "Newman", rather than the correct "Noyman" at 3:02.
bro dropping an enlightening video likes it's nothing 🔥
A brilliant and clear explanation. However, I still do not see how the BT paradox is not a proof by contradiction of the AOC and different formulations are need for well ordering etc ?
Small nit: At 19:30 the dashes for the itemized list can easily be read as negation signs, which is a bit confusing.
I liked this
Can we build maths on the category theory instead of set theory?
Yes we can, you can read “joy of cats”
This is duality;
Injective is dual to surjective synthesizes bijection or isomorphism.
Absolute truth is dual to relative truth -- Hume's fork.
Syntax is dual to semantics -- languages or communication.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory.
If mathematics is a language then it is dual.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are dual!
The integers are self dual as they are their own conjugates.
"Always two there are" -- Yoda.
Subgroups are dual to subfields -- the Galois correspondence.
@@hyperduality2838🤓
Now do homotopy type theory please
@32:00 lovely closing outro. But if these are "truths" they are not necessarily truths about "our universe" --- if by that you mean physical spacetime. The V of mathematics is different to the U of physics. Probably. ;-) If so, mathematics has this power to take a mind beyond physics. At least I've always thought so. (Banach-Tarski is not about physical bolls).
One of the deeper problems here is "logicism". Mathematics is richer than that and is a system _with which we we formalize_ basic numerical and geometric _intuitions._ But the intuitions are often thought to be "not mathematics". This is wrong, and yet true in one sense: our intuitions are not _formal mathematics._ One intuition is that if we throw a (mathematical point) dart at the line ℝ it will hit a real number (or a hyperreal that has a real _standard part_ ). Mixing such physics/geometric notions with formal mathematics is considered by math snobs and level-2 nerds to be a terrible "no no" but that's the problem. Logicism or formality restricts axiomatic development and gives rise to some of the unintuitive results. But the restrictions are a good thing, they give more creative mathematicians room.
Creating better formal mathematics out of basic intuitions is the task. No one said it could be perfected.
11:18 by the way, you can only prove single case by induction. if you want to prove statement in general, you need axiom of induction (:
You can prove induction. It is equivalent to showing that N is well ordered under inclusion. You can prove this from ZF.
@@ethanbottomley-mason8447 suppose you have statement S(n) for fixed n. You can prove S(n) without induction for any specified n. But you can't prove statement: "S(n) holds for all n" without axiom of induction. This is as far as I know.
@@r75shell Let A be the subset of th natural numbers where n in A iff S(n) holds not hold. Since N is well-ordered, then A contains a least element. You know S(1) is true (the base case), so let k be the least element of A, then k > 1, so k-1 is a natural number. Since k-1 < k, then k-1 is not in A so S(k-1) holds. You know that S(n) => S(n+1), therefore S(k-1) holds so S(k) holds. This contradicts the fact that k in A. Therefore A must be empty, so S(n) holds for all n.
This is duality;
Injective is dual to surjective synthesizes bijection or isomorphism.
Absolute truth is dual to relative truth -- Hume's fork.
Syntax is dual to semantics -- languages or communication.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory.
If mathematics is a language then it is dual.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are dual!
The integers are self dual as they are their own conjugates.
"Always two there are" -- Yoda.
Subgroups are dual to subfields -- the Galois correspondence.
Does the definition of bijective functions guarantee that all bijective functions are monotonic?
Can someone please explain more about why every chain in A ( 21:50 ) has an upper bound?
Given a chain {f_i} where i are in some index set S, we can define an upper bound on this chain. Let Y be the union of dom(f_i) over all i. For any x in Y, x in Y so x in dom(f_i) for some i, so let f(x) = f_i(x). Then f is an upper bound on the chain. It might look like we are using the axiom of choice to construct f, but in reality, understanding functions as relations, f is in fact just the union of f_i over all i in S.
@@ethanbottomley-mason8447 Thanks for the explanation. Allow me to reword the idea. The partial ordered of A is induced by the set inclusion relation, therefore we must treat the function as a set of ordered pairs to make the set inclusion relation to work. We say f_1 is smaller than f_2 if 1) the domain of f_1 is a subset of f_2 and 2) f_2 gives give the same output with f_1 while the input is in the domain of f_1. Then the upper bound of a chain will be the function which domain is the union of all function in the chain. The output has no ambiguity since we require that while the domain of two function overlaps, their output must be the same.
@@yuan-jiafan9998 That is correct. This is a fairly common technique for finding upper bounds when dealing with functions. You can use a similar idea to show that flasque sheaves have 0 sheaf cohomology.
everyone should try to define a well ordering on the reals or find a set with kardinality strictly between the naturals and the reals every now and then.
❤❤
I don't get why Banach-Tarski is counterintuitive. No one bats an eye when |(0,1)|=|(0,2)| cardinality-wise but when a sphere performs mitosis it's too hot to be plausible
I don't know for sure why, but here's a perspective.
How do we find the volume of compound geometric objects? Imagine a silo-shaped geometric object: a cylinder with a hemisphere on top. How do you find the volume? You split the object into two pieces (the cylinder and the hemisphere), find the volumes of those pieces separately, and then you add those volumes together.
Now, when we go to the Banach-Tarksi process, we split a sphere into 5 pieces, perform volume-preserving manipulations to those 5 pieces, and then we end up with 2 pieces, each having the same volume as what we started with. Viewing this process from a high-level, it brings into question our strategy of "break into pieces, find the volume of the pieces, and add those volumes together." If it doesn't work in the context of Banach-Tarksi, how do we know it's actually valid in other contexts?
There is a simple resolution to this fear: the 5 pieces we broke the sphere into don't all have a well-defined volume. There is at least one piece which does not have a defined volume (this is not saying the volume is 0; a volume of 0 would be a defined volume). So as long as we break our geometric object into pieces which all have defined volumes, then our strategy will work.
when you turn (0,1) into (0,2), you would most likely stretch it out by a factor of two. but this is not an allowed transformation and is not needed in the BT paradox.
Which statement is correct: 1., The set of sets that do not contain itself or 2., The set of sets that do not contain themselves or 3., The set of sets that does not contain itself? Once I get my bearings right in English grammatical rules then I may have a fighting chance of making sense of the ZFC😢
2
Suppose we develop mathematics without AC and use that to develop, describe physics. Would that work? What would quantum physics GR or Maxwell look like?
we can even go further. all of applied mathematics (the stuff running on our computers) don't only not need AC, they also don't need the axiom of the excluded middle. removing the two is called constructive mathematics
This is duality;
Injective is dual to surjective synthesizes bijection or isomorphism.
Absolute truth is dual to relative truth -- Hume's fork.
Syntax is dual to semantics -- languages or communication.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory.
If mathematics is a language then it is dual.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are dual!
The integers are self dual as they are their own conjugates.
"Always two there are" -- Yoda.
Subgroups are dual to subfields -- the Galois correspondence.
@@yonaoisme This is utter nonsense. Physics absolutely needs AC. It's implied from Krein Milman + Banach Alaoglu, both of which it needs.
Wait a minute... if I prove that AC existence of basis in every vector space, then I'm giving up hope of ever finding a vector space without a basis: if I accept AC, then I reject baseless vector spaces. But if I don't accept AC, then AC immediately becomes undecidable in ZF, I think? Which means I cannot find a baseless vector space anyway, since otherwise AC would become decidable.
you're on the way of becoming a constructive maths advocate
@@yonaoisme D: is it dangerous?
omg it looks sound to me
there is a set of numbers which exist on computers. It is quite large finitely, yet far from a complete representation of numbers, and the math that works on them may be a subset of pure math number concept.
Subset of numbers available on computers depends on the computer and what you mean by computer
The computer can or can not be a Turing complete machine, and for hardware specifics it can depend on the mathematical basis used and what functions are implemented, amount of memory etc
This is duality;
Injective is dual to surjective synthesizes bijection or isomorphism.
Absolute truth is dual to relative truth -- Hume's fork.
Syntax is dual to semantics -- languages or communication.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory.
If mathematics is a language then it is dual.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are dual!
The integers are self dual as they are their own conjugates.
"Always two there are" -- Yoda.
Subgroups are dual to subfields -- the Galois correspondence.
Does it work for the set of all complex numbers?
What exactly do you want to know if it works? The set of complex numbers is just R ×R with appropriately defined + and * operators (the pair [a,b] is by convention written as a+b*i, and the complex numbers are added and multiplied as polynomials with i*i=-1). It can be shown that R ×R can be mapped one-to-one with R; the same is true for R^n for any natural number n - set of all n-tuples of real numbers for some finite n, as well as R^N - set of all infinite sequences of real numbers. Likewise, N^N and 2^N (set of all infinite sequences of natural numbers, or of digits 0 and 1) can be mapped one-to-one with R.
@@MikeRosoftJH i have low pressure and cant understand
@@mihaleben6051 And I don't know what exactly you're asking. What do you want to know if it works with complex numbers?
@@MikeRosoftJH nevermind. I dont need to know anymore.
I’m the algorithm’s bitch because it just keeps recommending me exactly the videos I need, just like this one lol
Axiom of regularity: “any set contains a member that it is disjointed with” . I don’t get it - what does this mean?
This is duality;
Injective is dual to surjective synthesizes bijection or isomorphism.
Absolute truth is dual to relative truth -- Hume's fork.
Syntax is dual to semantics -- languages or communication.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory.
If mathematics is a language then it is dual.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are dual!
The integers are self dual as they are their own conjugates.
"Always two there are" -- Yoda.
Subgroups are dual to subfields -- the Galois correspondence.
Axiom of regularity says that every non-empty set has an element which is disjoint with it. (In conventional set theory - ZF or ZFC - all elements are themselves sets.) So let's see: what kinds of sets are there? Empty set satisfies the axiom by default. What about {0}? The sets 0 and {0} indeed are disjoint, because 0 has no elements, and so the two sets don't share an element. How about {{0}}? The only element of {{0}} is {0}. The only element of {0} is 0. So the two sets don't share an element, because 0 and {0} aren't the same set (they don't have the same elements: 0 is an element of {0}, but not of 0). (From the set theory axioms it follows that for any set A there exists the set {A} - set whose only element is A; and for any sets A, B there exists the set {A,B} - set whose elements are precisely the sets A and B. Note: element and subset relation is not the same thing.)
From axiom of regularity it follows that no set contains itself (or its superset) as an element. Likewise, you can't have sets A and B such that A∈B and B∈A. More generally, there can't exist an infinite descending sequence of sets: A, B, C, D, ..., where B∈A, C∈B, D∈C, and so on. (Take the set of all sets in the sequence, it can be seen that none of the sets A, B, C, ... is disjoint with this set.)
At 13:26 you wrote f instead of g for your injective function.
This is duality;
Injective is dual to surjective synthesizes bijection or isomorphism.
Absolute truth is dual to relative truth -- Hume's fork.
Syntax is dual to semantics -- languages or communication.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory.
If mathematics is a language then it is dual.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are dual!
The integers are self dual as they are their own conjugates.
"Always two there are" -- Yoda.
Subgroups are dual to subfields -- the Galois correspondence.
You forgot to mention the Axiom of Determinacy.
Bretty gud.
Was the Axiom of Choice a redundant constraint just to counter the dogmatic determinacy dictated by the Monotheistic religious beliefs, as such the AC being more of a political statement than a real Axiom? Once this choice was made, however of course one can defend it as necessary or refute it. I guess the ambiguity can be dissolved by a different theory entirely, such as the Homotopy Type Theory or other theories that favour the absence of infinities of whatever cardinality. I would love to find a consistent and robust answer to all of these assumptions. Or is this the realm of Godel's theorem?
5:37 okay I'm completely lost
Really needed this about 4 months ago
Sets feel like "who's on first." The math that I don't like is where physicists think math can dictate cause to them. To get a physics degree, everything is 100% math. Cause is an afterthought scoffed as "the intuitive explanation," and yet I can prove that dancing around math analogies can hide the real cause from you. "The wheels on the bus go round and round." This song describes EVERYTHING you see a bus do exactly like math does, but it is not an understanding of a bus. Modern science WILL NOT see past the FIRST successful description. Constants are SET in stone. You think angular momentum causes the gyroscopic effect, and it is well known that angular momentum is a fake vector. Make sense of that one. I have a video that shows what cause actually looks like, and it's based in absolutes.
Mathematics
It's a bit innacurate to describe transfinite ordinals as "larger" than countable infinite. Most (maybe all?) ordinals still have cardinality aleph null, so they are the same size. Successive transfinite ordinals are not successively larger, they just come after those proceding them.
"Most (maybe all?) ordinals still have cardinality aleph null, so they are the same size."
No. There are infinitely many ordinal numbers of each cardinality. And the rigorous way to define cardinal numbers in ZFC is as a special subclass of the ordinal numbers. And indeed, the hierarchy of transfinite cardinal numbers (assuming the axiom of choice) can be defined exactly as the video describes. Fix any transfinite ordinal number α, and take the set of all ordinal numbers β which inject into α. This set has a supremum in the ordinal numbers, and that supremum will be an ordinal of the "next" cardinality after the cardinality of α.
@@MuffinsAPlenty yep, I was just thinking of omega up to epsilon_0 etc.. Everything that can be reached from below with an "and so on" process. Epsilon_1 and up have cardinality aleph_1 or greater.
the ordinals are a vast superclass of the cardinal numbers. both are so huge that they are not actually constructible in ZFC. they are called classes
Wouldn't that imply that most sets are countable? which is obviously false, if anything countable sets are a small minority
@@fedem8229 yeah that's right, I was just thinking along the lines that ordinal sets up to epsilon_0 are described by a sequential process, any set that can be described as the limit of a sequential process will have countable cardinality.
Ngl, I thought the last 30 seconds were introducing a Skillshare sponsorship
Gödel
Mensura nihil aliud est quam id quo quantitas rei cognoscitur : quantitas vero rei cognoscitur per unum et numerum. Per unum quidem sicut cum dicimus unum stadium vel unum pedem ; per numerum autem sicut cum dicimus tria stadia vel tres pedes ; ulterius autem omnis numerus cognoscitur per unum, eo quod unitas aliquoties sumpta quemlibet numerum reddit '. St. Thomas, In X Metaph. lect. 2.
english translation (google):
"A measure is nothing else than that by which the quantity of a thing is known: but the quantity of a thing is known by one and number. By one indeed, as when we say one stade or one foot; but by number, as when we say three stadia or three feet; and further, every number is known by one, inasmuch as unity, taken several times, gives any number. St. Thomas, In X Metaph. read 2."
Omg
What level of mathematics is this?
Basic university level
I would say advanced undergrad if you want to get deep into the details, it's not something you would usually cover in the first 2 years as a math major, and it's (sadly) optional at my university
@@fedem8229really? I'm a comp sci major and I had all of this in my first year. That's interesting
Was about to say how great this video was, but then saw you include 0 in the natural numbers.
What?
I kinda, don't buy the idea that just because sets can describe everything, it makes, them fundamental. Language can also describe everything, that doesn't make it fundamental and Math can exist without it.
This is duality;
Injective is dual to surjective synthesizes bijection or isomorphism.
Absolute truth is dual to relative truth -- Hume's fork.
Syntax is dual to semantics -- languages or communication.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory.
If mathematics is a language then it is dual.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are dual!
The integers are self dual as they are their own conjugates.
"Always two there are" -- Yoda.
Subgroups are dual to subfields -- the Galois correspondence.
Language can only describe a countable number of things, not everything.
Yo I wrote this to define Axiomatic system "The Axiomatic system that is completely described is a special kind of formal system that transcends the philosophy of both mathematic, and logic, that within necessary entailments will ensure expressive Axiomatic properties inquiring to critical theorem conclusive of reality"
This is duality;
Injective is dual to surjective synthesizes bijection or isomorphism.
Absolute truth is dual to relative truth -- Hume's fork.
Syntax is dual to semantics -- languages or communication.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory.
If mathematics is a language then it is dual.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are dual!
The integers are self dual as they are their own conjugates.
"Always two there are" -- Yoda.
Subgroups are dual to subfields -- the Galois correspondence.
You cast axioms as constructs on top of mathematical objects. But that's just not true. The word means things you simply declare true. Similar to a = 4 but general and so fundamental. Take numbers themselves. You start with a few axiomatic declarations and you build sets upon them. The theory of non-contradiction, the excluded middle - they're all axiomatic.
Say I want to construct a 4d space with my computer. I declare 4d vectors, I decide if it's linear, I insert an orthonormal basis. When this is done, I take these axioms and I use the logical axioms to build upon them until the person running the programme, projected onto the 2d screen is very confused.
It's a starting point, not an ending. It's not a conclusion, it's an assumption.
Correct, axioms are statements that are either accepted as true upon the basis of no evidence whatsoever, or are rejected upon that same basis. In fact, evidence isn't permitted one way or another. Axioms must simply be accepted or rejected, they have no other function.
This is duality;
Injective is dual to surjective synthesizes bijection or isomorphism.
Absolute truth is dual to relative truth -- Hume's fork.
Syntax is dual to semantics -- languages or communication.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory.
If mathematics is a language then it is dual.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are dual!
The integers are self dual as they are their own conjugates.
"Always two there are" -- Yoda.
Subgroups are dual to subfields -- the Galois correspondence.
@@hyperduality2838 Weirdest comment I've read in ages, reads like drug addled new-age word salad and yet, yeah, Hume's Fork genuinely applies. The "analytic-synthetic distinction" is relevant drawing a distinction between things we simply define to be true and those we conclude as likely to be true by observing reality. It's the reason why maths has proofs and science has theories.
@@davidmurphy563 Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Sinh is dual to cosh -- hyperbolic functions.
Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases or Riemann geometry is dual.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non teleological physics (entropy).
Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
Positive is dual to negative -- electric charge or numbers are dual and curvature is dual.
Duality creates reality.
Addition is dual to subtraction (additive inverses) -- abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- abstract algebra.
Integration (summations, syntropy) is dual to differentiation (differences, entropy) -- abstract algebra.
All numbers are dual if they fall within the complex plane as complex numbers are dual.
Real (kinetic energy) is dual to imaginary (potential energy) -- energy is dual!
Analytic, rational is dual to synthetic, empirical -- Immanuel Kant.
Action (thesis) is dual to reaction (anti-thesis) creates the converging or syntropic thesis, synthesis -- the time independent Hegelian dialectic or Sir Isaac Newton.
Forces are dual -- attraction is dual to repulsion.
There are new laws of physics.
@@hyperduality2838 My balls are dual; so what? Baryons are triple. Forces are quadruple. Anyway, sine and cosine aren't duals - there's also tan. They're ratios in a right-angle triangle; so three, not two.
How is a vector in column space and a co-vector in row space "dual" in any meaningful way?! Okay, let's give you the benefit of the doubt. Let's use a specific example:
Say you have a vector and a co-vector matrix which is tangent to a manifold. Let's say we have a vector v at point P. Let's make theta = 0 and phi = PI/4 and we'll use spherical coordinates where v corresponds to {0, 1}. Let's make a co-vector v2 at P where v20 = 1 and v21 = 0. In matrix form {0, 1}. Simplest example I can think of and 100% in Riemann space.
How does the word "dual" meaningfully apply to that in any way? If you want to show me you're not just using buzzwords, feel free to show me a proof by contradiction from the above using the inner product.
Because I have a feeling you have no clue how to do that or what any of these words actually mean.
At 26:16 you say "a bit more easier". I'm sorry to nit-pick, but 'easier' is already a comparative so 'more' is redundant. But this tiny error hardly detracts from the video, of course.
🤓🤓🤓
Woah woah woah. No. Banach Tarski is implied by *_Hahn Banach._* It's strictly weaker than Choice. And you can't do Functional Analysis without Hahn Banach. It's a fool's errand. Banach Tarski is a demon that cannot be exorcised.
Cardinal magnitude is a canard. What we must ultimately understand is the (time/distance) Cartesian relationship and the nature of number that is implicitly bound to time. What we're really looking at is a rate of growth. Iteration along a 1 dimensional number line is sub-unitary. The natural base, e, is the unitary rate of growth in the universe. The concept of power sets, much like the concept of a Fibonatchi based rate of growth that we see in natural processes, is an algorithmic based transform on a given set with a certain growth curve. The idea that we have these "collections" of objects - integers say - in some bag and we can reach into our set/container/bag and simply withdraw either the first element or the infinity minus one element with equal speed, precision, and ease of efficiency is not supported by actual calculations, computer systems, or any extant form of symbolic representation.
Space is dual to time -- Einstein.
This is duality;
Injective is dual to surjective synthesizes bijection or isomorphism.
Absolute truth is dual to relative truth -- Hume's fork.
Syntax is dual to semantics -- languages or communication.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory.
If mathematics is a language then it is dual.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are dual!
The integers are self dual as they are their own conjugates.
"Always two there are" -- Yoda.
Subgroups are dual to subfields -- the Galois correspondence.
what a great video, amazing set theory propaganda :)
huh? just because he mentioned "every ring has a maximal ideal"?
@@yonaoismethe video literally starts with an explanation of the importance of sets in mathematics and the set theory axioms
@@mushykitten my guy, your original comment was "algebra propaganda"
@@yonaoisme I can also make up stuff
@@mushykitten you know that everyone can see that your comment is edited, right?
🗝️🪐
Banach-Tarski has bothered me a lot less since I realized that, given that planar construction geometry as per Euclid, only points with rational coordinates are in the constructible plane. So there are uncountably infinitely many holes in a continuous plane already. What's a few more? I suppose it would take a lot of work to try to figure out just how many intuitive and even axiomatic assertions there are because the Euclidean plane is quite literally nothing compared to a continuous plane.
Minor nitpick: square roots of constructible numbers are constructible, so there are some points with irrational coordinates which are constructible, e.g., (sqrt(2),sqrt(2)). Nevertheless, your main point stands: it is still true that there are only countably many constructible points on the plane. This follows from the fact that all constructible numbers are algebraic and there are only countably many algebraic numbers.
@@MuffinsAPlenty Yes; you're quite right. Still, there are either ℵ₁ or ℵ₂ holes, depending on what you prefer. I like CH because it lets me sort of visualize up to ℵ₄ on a good day. I leave the idea of whether this has anything to say about classical versus quantum physics to the reader.
do you have sleep apnea
who is the audience for this? i can’t really see how this would teach anybody who wasn’t already familiar with these topics anything. Within minutes of explaining what a set is you’re already busting out set theory jargon with very few examples a layman would understand.
Arithmetic cannot be completed (Incompleteness), but it can only be described robustly within multiplication.
Unlike Addition, Multiplication requires its Elements to be identical for validity.
Under Multiplication the Integers are a finite, but unending population of slopes, no longer fully infinite.
More like ASS-umption of choice.
Show me an example of an infinite set. Showing the first few numbers and then three little dots is not the same thing.
the set of all whole numbers
Consider the set N with the following properties:
i) 0 is in N
ii) if m is in N, then m+1 is also in N.
Such a set exists in ZF and it is in fact not finite
@@jussari7960 But of course, you can't prove that this set exists except by taking as an axiom that it exists.
@@MikeRosoftJHYeah, if you dont like it, you can of course choose to not use AoI, and you'll get some form of finitism. It's just a lot weaker than ZF(C) so most mathematicians do assume AoI
@@jussari7960 I believe that theory of finite sets has the same strength as first-order Peano arithmetic.
I don't like the axiom of infinity.
I don't think we're allowed to dislike any of the axioms. I mean, we _can,_ but mom will get mad at us if we do.
Abstract algebra not only contain meaningful abstractions and meaningless ones, so in conclusion not only they are undecidable but also fail to arrive at meaningful objects. As for other branches of mathematics the theorems of abstract algebra doesn't apply.
why does the chin look like tha
Can't you at least spell-check your slides? I am only a few minutes in and I have already seen "Russel's" and "Bernsays". I have also heard you mispronounce "von Neumann" as "von Newman". The technical content is OK so far, but these other issues are irritating.
Why so snarky about non-relevant mistakes when being kind is free
Are you similarly unpleasant in real life?
This is duality;
Injective is dual to surjective synthesizes bijection or isomorphism.
Absolute truth is dual to relative truth -- Hume's fork.
Syntax is dual to semantics -- languages or communication.
Categories (form, syntax, objects) are dual to sets (substance, semantics, subjects) -- category theory.
If mathematics is a language then it is dual.
All numbers fall within the complex plane.
Real is dual to imaginary -- complex numbers are dual.
All numbers are dual!
The integers are self dual as they are their own conjugates.
"Always two there are" -- Yoda.
Subgroups are dual to subfields -- the Galois correspondence.
@@hyperduality2838 I recognise you. You have seen one half.
@@Juttutin Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Sinh is dual to cosh -- hyperbolic functions.
Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases or Riemann geometry is dual.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non teleological physics (entropy).
Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
Positive is dual to negative -- electric charge or numbers are dual and curvature is dual.
Duality creates reality.
Addition is dual to subtraction (additive inverses) -- abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- abstract algebra.
Integration (summations, syntropy) is dual to differentiation (differences, entropy) -- abstract algebra.
All numbers are dual if they fall within the complex plane as complex numbers are dual.
Real (kinetic energy) is dual to imaginary (potential energy) -- energy is dual!
Axioms are not “true”. This is a meaningless phrase. Axioms are just a assumed so that the logical consequences of that assumption can be explored
bro thinks hes 3Blue1Brown😹😹😹
Dude this is fucking awesome man! Keep it up
14:32
The domain of h in this case doesn't have to be P(X)\{∅} right? It can just be the union of all Xy such that y belongs in Y.