well, the peano axioms demonstrate that numbers are not really specific entities. They are more of a property of things within some system, be it sets, apples, money, bits, energy,... as long as the system satisfies peano axioms, the things within that systems are numbers.
what I thought. I get the feeling he went on a whole rollercoaster ride of critical thinking without regarding the simple fact that the concept of number is a reference to/by-product of naturally occurring quantities found throughout the universe
Well, he does bring up a point at 2:04 that mathematicians of the XIX and XX centuries were very uncomfortable with this vagueness, and wanted to put all of numberhood on firm, solid footing.
The axioms go a bit beyond what is usually considered an interface in say Java. In Java an interface for defining the class "WholeNumber" would declare a defined constant "ZERO" and a function "WholeNumber successor(WholeNumber )", But there is no way to describe any constraints on how successor() and ZERO behave. For example, you can't say in a Java interface that ZERO cannot be the output of successor(x) no matter what x is.
what if you were to introduce the concept of type traits and constraints (thinking meta c++ here) into the mix to define what (types) the results of S (and x (and ZERO)) are - the type manifests the briefcase/doll concept, so each different type in the system represents a different number..
I think you are overthinking; it's just an analogy :) It's pretty clear that programming languages have problems (by design) when representing structures without actual concerns about computability and resource handling, and Mathematics is purely structure. As a side note, I know that some cathegory theorists use an "extended Haskell language" sometimes as an alternative to abstract algebra and arrows diagrams; it's a cool language to express mathematical ideas algebraically.
This could be my favourite episode so far. Great topic, interesting, deep and even philosophical, but set up in a way that makes it accessible to pretty much everyone with an interest in this kind of thing. Great job! You should definitely make more of these - the really technical stuff, although interesting, isn't always accessible to many of us still busy with our undergraduate degrees : (
If someone wants to understand 9:27, you can try to prove the following: Every countable set can be equipped with a function S so that it satisfies the peano axioms. Even more exact: If you have a bijective function f from the natural numbers to a set X, then there is exactly one function S' on X so that S' composed with f is the same as f composed with S, and S' turns X into "a set of natural numbers" (fullfills the peano axioms)
And in the set theory ECTS we says such a set exists, full stop. No need to construct them from any some "primordial" soup like von Neumann/Zermelo constructions
So interesting! This is the first time I've heard a lot of these concepts, but I'm always impressed at how well you explain them. What a fascinating way to think about numbers! Thank you!!
Gabe, I love the personal wisdom you provide at the end of the video. Regarding which model is preferred, I think the VN model is beautiful for its ability to convey arithmetic in simple set operations. The less than as a subset you mentioned already. Addition in VN is the size of the disjoint union. But of course we have to be careful saying size so we get around that by saying that we want the "pure set" that is isomorphic to the disjoint union one that is all fuddled with duplicates. Thanks for another wonderful video Gabe. Have a great weekend!
When algebraic structures are generalized they become more powerful and you can find more mathematical objects that behave like the original structure you abstracted. Examples: vector spaces, groups, rings, fields... So, when the structure of natural numbers is abstracted, is natural that somente will find other things that behave like them. That's awesome and beautiful.
The object of mathematics is not numbers but mathematical language. Or, to state it in linguistics terms, math is a metalanguage that operates on its own concepts (containers). It so happens that the containers/concepts of math are paradoxically the 'content' of Nature. So mathematical language seeks to be the projection (or subsumption) of Nature into an intensional set of concepts.
I've had a very interesting convo with a prof. He suggested that on any infinite set X (so there are injective functions onto itself that are not surjective, take some such F). You take an element of the set outside the image of F, and take the chain of that element. It's easy to show that the set we get from the chain (name it C) and some operation "+" -- recursively defined by F restricted to C -- form a commutative monoid isomorphic to ℕ. Interestingly enough, you can get some ordinal structure by finding other elements outside of F's image and "glueing" those chains together. You get a weird order where every element outside of F's image is made into a ''limit ordinal''.
@Gabe, PDF contains error. emailed ya. "30214/143" is missing the operator "+" between "302" and "14", to provide the correct interval between valid clock configurations. alternately, change "30214" (the incorrect number of seconds in a day) to "43200", and this is also a valid fix. will keep an eye out for anything else.
+PBS Infinite Series Since you've done videos on sets, foundations, functors, topology, and logic, would you ever consider doing a video on Topos theory?
Really like this episode and the previous one. Thank you for making these videos! My personal view aligns with yours. It is meaningless to ask the meaning of anything in mathematics, all these objects emerges one way or another from some fundamentally irreducible assumptions.
The suitcase concept reminds me of blockchain, in the sense that each new block is built on the hash of all previous blocks. The lesson would even synergize with the videos on cryptography and hashing. Edit** PBS please do a video on blockchain!!
I suppose it's fitting that in set theory, everything - starting with numbers - is made from (you guessed it) sets. I like that fact that the empty set is 0, then the next set (1) contains 0... just like computers start counting at 0, rather than one. I prefer the Reese's Peanut Butter Cup Theory of Everything, where 0 is an empty wrapper... tavi.
If 3 is really the same as {0; 1; 2} it would mean that you should be able to iterate over numbers in a for loop in programming! In Python, if you want to iterate over 3, you'll have to write "for i in range(3)". In any other language, you'll have to write "for(i=0;i
Great video! I think it would have been nice to put more emphasis in distinguishing in the beginning ordinal numbers and cardinal numbers. While watching the video I kept thinking to myself that the numbers you define do not "feel" like numbers because I was thinking about cardinals.
Great episodes, love this fundamental stuff! I have a request for an episode on Grassmann numbers, the exterior algebra, the Berezin integral etc. Maybe in cooperation with PBS spacetime on supersymmetry? :) Please :) Thanks! :)
at 9:50 you mention there being multiple ways to construct R out of N. I've only ever seen the Dedekind cut version presented previously on this channel. I'd like to see an episode with more detail on some of those ways.
I see a distinct rise in quality of this channel's content. Nice concepts covered with little nuance (and if there is it's mentioned for people to go look up). Keep up the good work.
Well, the interesting question is asked in the second half of the video - anyone can see upon further reflection that these set-theoretic constructions formally parallel operations with numbers when the right relations are specified, but this doesn't of itself show that what a number is is a set. It especially does not show that 1-as-such is an operation in which 'nothing' is taken to be an object of the intellect (which seems to be Cantor's conception of what a set is). At least intuitively, this is not what what we mean when we say that there is _one_ thing - we do not take the empty set (0 in this formalism) to be an object of the intellect and conjoin that with the quantity of that thing. In a similar way, when I say there are two things, I do not mean "take now as an object of the intellect both 'nothing' (0) and another object of the intellect which is in turn 'nothing' (1)". This set-theoretic construction has always seemed to me to be formalism (a position in philosophy of mathematics) gone wild and not saying more essential to number-as-such - I think the Peano axioms do tell us something, but not set-theoretic constructions like these. But, when it comes to math, I'm a filthy Platonist (realist), so my solution for now is to say that the essence of number is that which is in common, the form, of all these constructions, with individual constructions being different names of the same form; this is what I take to be the insight that these Peano systems are all isomorphic to one another and what the general importance of proving two things to be isomorphic in math is.
If you expected that the construction would tell you anything more about the Peano Axioms or Peano Arithmetic, of course it won't. It is a construction, a demonstration that the set N used in the Peano Axioms can in fact be described (if you're a Platonist you should consider that important). Some mathematicians are constructivists and demand a realization of an argument before accepting it.
Hi, thanks for these videos, they were great. But I have some questions: 1)How we define "function" when we construct natural numbers using Peano axioms, for example, can the notion of function itself be constructed in terms of sets ? 2)You sad that other formulation of the Von Neumann model is "A natural number is a set X, each element of which is also a subset of X". Maybe I misunderstood something, but don't any set satisfy this condition ? 3)And last, in the first video you sad that "=" is any relation that is reflexive, symmetric and transitive. But then what means for example that function of some input equals to some output, in the case of the video, what means S(Zelda)=x ? (I suppose that this question is closely connected to my first question). Thanks in advance. And thanks for the videos.
Great questions! 1) Yes, a function can be constructed just through sets, but it's a bit tricky. First, we need to define the notion of a "tuple", meaning an ordered pair of _things_ (not saying "numbers" here on purpose, as it could be anything). If you have the tuple (A,B), how would you define that as a set? Well, you could do {A, B}, but that loses the order, as that would also be the representation of (B,A). Instead, the definition of a tuple is {A, {A,B}}. The representation of (B,A) would be {B, {A,B}} which is different. Now, you can ask yourself: What even is a "function"? Well, it's something that maps objects from one set S to objects from a set T. But every mapping from one thing in S to another in T is just a tuple (A,B) where A is in S, and B is in T. So, a function is just a set of tuples {(A1,B1), (A2,B2), (A3, B3),...} where the A's are in S, and the B's are in T. Additionally, all the A's need to be distinct, as you can't map the same A to different B's. If the tuple (X,Y) is an _element_ in the function f (because, as stated, f is just a set of tuples), then we write f(X)=Y. What that equals-sign means is answered in 3). 2) Not quite. For example, take the set "{Dog, Cat, Mouse}". The elements of it are "Dog", "Cat", and "Mouse". However, none of these elements are a set at all, so they're not subsets of the original set. "{Dog}" is a subset of "{Dog, Cat, Mouse}", but that's not the same as "Dog". Even if "Dog" is a set, it is certainly different from the set "{Dog}" which contains just one element. It's the difference between having a dog with you, and carrying that dog in a suitcase. In the VN model, the number 4, for example, is just {0,1,2,3}. However, the number 2 is just {0,1} which is a subset of 4. 3) He glossed over what he meant by "relation", but it's also just another set of tuples, except with even less restrictions than functions. In fact, a function is just a specific _type_ of relation. The set {(A,B),(A,C),(B,C)} is an entirely legitimate relation, but not a function because A is "mapped" to both B and C. In this case, it's transitive because (A,B) is in it (meaning A = B), (B,C) is in it (meaning B=C), and (A,C) is in it (meaning A=C). It is neither reflexive nor symmetric, though. For a relation to be reflexive, it must contain the tuple (X,X) for any element X. For a relation to be symmetric, it must contain (Y,X) if it contains (X,Y) for any elements X,Y. I mean it when I call those great questions, by the way. Because it takes a keen understanding of mathematical thinking to notice that Gabe left some things undefined, which at first glance seem obvious. That's the sort of thinking which got us to the Peano axioms in the first place.
Yes functions can be created using only set theory and a system of logic. A function is simply a set of ordered pair where the first element is the "input" and the second element is the "output" (actually this is a more general concept called a relation, you need to add some restricts to get a function specifically). And yes ordered pairs can be defined in set theory too but doing so is both incredibly difficult and largely pointless.
1) Yes it can. A function F from set A to B is a subset of AxB. So every element of F is a tuple (x,y) with x∈A and y∈B. That makes F a so called "relation". In order to make F a function, for every x∈A there has to exist a y∈B such that (x,y)∈F. Furthermore that y has to be unique. There you go. A function from A={1,2} to B={a,b,c} could look like this F={(1,a) , (2,c)} and you can change to the common notation F(1):=a, F(2):=c. F={(1,a) , (1,b) , (2,c)} would not be a function, because the image of 1 is not unique. You would say "F is not well defined". 2) No, that is not true for every set. Take this set A={{x,y},z}. {x,y} is an *element* of A, but *not* a subset. In order for {x,y} to be a subset of A, it would require that x and y are elements of A, but A has only two elements namely {x,y} and z. 3) Like in the first point a (binary) relation R is defined as a subset of a cross product of two sets A and B (in the case of an equivalence relation, A and B have to be the same set). So R⊆AxB. What we commonly mean by "=" is a so called equivalence relation which is again just a relation with certain properties, namely reflexive: If x∈A then (x,x)∈R symmetric: If (x,y)∈R then (y,x)∈R transitive: If (x,y),(y,z)∈R then (x,z)∈R S(Z)=x is just another notation to essentially say that the tuple (S(Z), x) is element of that equivalence relation we call "=", where S(Z) is the unique element y such that (Z,y)∈S. edit: I notice that I just assumed it is okay to use tuples, when we only have sets. However tuples which are nothing more than ordered sets can be constructed by using unordered sets. One way to do it is (x,y) to be defined as {x,{x,y}} and voila, if I give you the set {{3,2},2} you immediately know that I mean the tuple (2,3) and not (3,2). All these constructions using only sets can get really confusing, but it is also funny to think that you can rewrite something like 2=1+1 as {2,{2,+(1,1)}}∈= and even more confusing as as {{{},{{}}},{{{},{{}}},+({{}},{{}})}}∈=
In every academic setting in which I have been, if "function" is being formally defined, it has been defined as Treufuß did. Nevertheless, I confused an entire classroom of my fellow students when I described a function between two small sets by listing the tuples.
You mentioned how you can construct the projective plane as a quotient space of a square. The frustrating thing about this construction is that it is not smooth at the point corresponding to the corners of the square, unlike the torus and Klein bottle constructions. If the Pacman board was a projective plane instead of a torus, and Pacman runs directly into a corner, the direction in which he leaves the corner is indeterminate.
It somehow feels that boolean values would be a good stepping stone between natural numbers and nothing. Do they have any part in the world of number sets?
At 7:26 he says about the Von Neumann construction: "Every number is a set each of whose elements is also a subset of the set". This is true for the Zermelo construction, right? Except that Zermelo only as the "predecessor" element as a subset, but Von Neumann has all predecessors as elements. I am wondering if I am missing a deeper significance of elements being subsets itself ... the neat thing is just that all predecessors are subsets?
No, it's the von Neumann construction. Every element of an element of a vN ordinal is also an element of that ordinal. For example, 3 = {0,1,2}, with 0 = {}, 1 = {0}, and 2 = {0,1} all being subsets of 3. If we take 3 = {2} = {{1}} = {{{}}}, then 2 is not a subset of 3.
Numbers for poly-semiotic observation by any neural network are made of quasi crystalline-- E8 recurrent-- time crystalline glyphs for neurAl network use and interpolation into ideAl symbols then iconic signifiers then axiomatic stabilizers (of toric code as seen in quantum computers).
That foundations of mathematics WAS developed in order to reduce the number of initial assumptions, that's not really a matter of debate. It's a historical fact. Higher mathematics encountered a bunch of problems caused by faulty assumptions. Mathematicians wanted to avoid that kind of thing.
Alexander F that is true, though Zermelo had a very different conception of what a set is than Cantor had, which is somewhat fascinating from a historical PoV. Check out the research article by F.W Lawvere on Cantors concept of Lauter Einsen.
However, modern interpretations do not in any way have to follow historical motivations (modern physics is plagued by this phenomenon). So there is still room for debate if one wants to, depending on how the question is phrased.
Re: "It doesn't matter what things in math really are", I certainly agree. Most of what we do in math relies on objects' properties rather than what they're "made of". Both of the constructions shown are no more than a few centuries old, but we've been using the natural numbers and proving things about them for millennia! The ancient Greeks used geometric constructions, but the exact same results with the exact same proofs still hold today. I'd even go so far as to say the properties are more fundamental, since we choose the constructions so that we have the properties we want. The Peano axioms were chosen so that they satisfied the intuition we already had about ℕ, and the set constructions were chosen to satisfy those. Not the other way around. To put it programming terms, we use the interface and only have the implementation for the sake of having one.
I get the feeling he went on a whole rollercoaster ride of critical thinking without regarding the simple fact that the concept of number is a reference to/by-product of naturally occurring quantities found throughout the universe
If zero is an empty collection. Then a more general question would be: what is memory? Memory is only an association between two "things", so it could be represented as a network. Numbers are then a special case of a network where all associations are sequential, having a beginning and an end because otherwise it would be a circle..
I would say that numbers as physical properties are made out of physical objects. But numbers as abstract objects are made out of constructions and axioms.
Something just popped into my mind while watching this and it was very interesting. Can you have an axiomatic system such that its axioms contradict one another? It might be a useless system but that's not the point? Or am I just being stupid?
I ca make a tuple theoretic model for the finite surreal numbers. That would be the Dyadics, which are the numbers you find on an old fashioned ruler with imperial divisions (fractions of inches).
For the 3 handed clock (hours, minutes, seconds) "swappable times" you'd be naturally limited to those times when one of the hands bisects the angle between the other two. Which hand that is in the middle will vary. I've no idea if there are any valid examples of this.
That's the von Neumann definition of natural numbers: any natural number is realized as a set of all numbers less than itself. So 0 is an empty set, 1 is {0} (set whose only element is an empty set), 2 is {0,1}, 3 is {0,1,2}, and so on. Observe that any natural number n is an n-element set. In addition, observe that natural number n+1 is obtained from n by adding n itself to it as a new element - n+1=n∪{n}. Under this definition, natural numbers are ordered by the ∈ relation; we can define a
Interestingly you had to define a function. I think the best way of defining natural numbers is in terms of functions themselves, like in lambda calculus. Please, make videos on category theory.
Regarding 9:07, doesn't that statement contradict the existence of non-standard models of Peano arithmetic? (Of which there are plenty by the Löwenheim-Skolem theorem.) I don't see how the standard model could be isomorphic to a non-standard one.
Löwenheim-Skolem is about 1st-order logic and would apply to a 1st-order formulation of the Peano axioms (and models thereof). But the version of the axiom of induction that I reference in this episode & in the previous episode "What Does It Mean to be a Number" is a 2nd-order statement. This is precisely the thing I said I didn't want to get into in my disclaimers at 3:21 (second bulleted item there), but the question you raise is valid. Short answer -- it's a 1st-vs-2nd- order issue, and a 2nd-order formulation of the Peano axioms is categorical (all models thereof are isomorphic).
@@pbsinfiniteseries But then you're going to make the set: "{0, 1, 2, ...} + something". Sure, second-order Peano arithmetic has an axiom: "given any subset of natural numbers, if 0 is in the set, and for any n it also contains S(n), then the set contains all natural numbers". So this set would seemingly not comply with the axiom, because you could take the set {0, 1, 2, ...}, which indeed contains 0 and for any element n it contains S(n), but this set does not contain all elements. But here's the problem: who said that {0, 1, 2, ...} were a set? (Second-order PA only has a unique model up to isomorphism within a particular model of set theory.)
I haven't watch this yet, or the previous episode. But I will with interest (even though it kind of goes over my head). I'm wondering if "cosmic black holes" are related? - I watched an introductory lecture by Leonard Susskind on black holes (I actually learnt stuff - the key things he wanted people to take away). One of things I took away from it (and I expect I have this wrong) is that it may well be that there is nothing inside a black hole? - not because the stuff inside the black hole "vanishes", but because of the breaking of entanglement at the "fundamental" level of information, i.e. like an entangled virtual partice pair forming near the event horizon of the black hole and splitting, for one particle to fall past the event horizon towards centre of the black hole and the other to escape, and in the process a photon is created that also leaves the black hole. The idea is the escaping particle loses entanglement with the particle inside the black hole, but entangles with the escaping photon instead. So information is conserved about the particle in the black hole - and by virtue of this, it doesn't need to exist and so, some how, it doesn't? (this is why Hawking Radiation causes black holes to loose mass?). Though at the same time the particle falling into the balck hole could exist as scrambled information on a holographic encrypted event horizon - though whether this consitutes "existing" if it is "fubar-ed" I don't know? I'm wondering if the same kind of vanishing trick applies to number sets on a fundamental level? - that you can never reach a "singularity" or "zero", but yet these concepts still logically and physically seem to still exist by virtue of what is around them. Like with the arrow of time, it seems like such things are probabilistic? Bip. Thanks.
Chris: So-called (spinless) black holes are just Schwarzschild solutions. They don't have an actual singularity and can be analytically extended well inside the Schwarzschild radius. A lot of misunderstanding comes from treating the distant, nearly flat background as applying "within" the black hole, whereas spacetime becomes extremely curved as you get "inside" it. If proper care is taken to identify the time direction, you can create an Einstein-Rosen bridge, aka wormhole, everywhere smoothly differentiable.
Doug Gwyn Thanks. I neglected to mention anything about general relativity in my comment didn't I :P ...I understand relativity pretty well - like a layperson's understanding.
The minimal construct of everything organized must be broken down to elements of thinking consistently. That just made me think, is there a minimal set of axioms that define a set of unintelligible sets? Maybe its a stupid question in that its only those things which are opposite of any logical construct. Like for example, any number in a set is not a set of itself. I'm no mathematician, but I love asking stupid questions :).
To get incoherent sets you don't need any axioms. Assume anything you can describe is a set. Now is there a set which is "the set that contains all sets"? There should be since we can describe it but then we run into paradoxes.
Michael Novak that's more like a definition (like the peano axioms) not a construction (like the von neumann construction). My favorite is probably one that involves equivalence classes of Cauchy sequences of rationals, but I also like the simplicity of the reals as finite surreals that are simply approximated by rationals.
I don't have one. I don't like reals at all, because they involve an uncountably infinite amount of incalculable, unknowable numbers. I think it would be fascinating if there would be a commonly known set of numbers that have some nicer properties. Defining numbers in terms of Turing machines that output Cauchy sequences would be one, but even then there's all kinds of problems with the halting problem, which leads to problems with undecidable equality and ordering between numbers...
+Pyry Jontio Agreed. There is no valid mathematical construction of the reals, especially the unconstructible numbers (a subset of the reals). You can get away with constructible numbers I'll give you that, but even that requires an a priori notion of a function which is a very vague object for building foundations on (even more vague than sets). But when it comes to Infinite choice, Cantor's infinities, that kind of math, yeah nah it's all bogus. You've been watching Wildberger haven't you :)
Assuming the Van Neumann construction is correct and 3 is really the same as {0; 1; 2}, what does a non-natural number look like? I am referring to negative numbers, rational numbers, and irrational numbers!
This is covered in the video "Crisis in the foundation of mathematics". Integers are a closure of integers under the operation of subtraction. So we define them as pairs of natural numbers [a,b] (by convention written as a-b), under the equivalence relation [a,b]~[c,d], if a+d=b+c. Rational numbers are a closure of integers under the operation of division by a non-zero numbers. So once again we define them as pairs of integers [a,b] (by convention written as a/b), under equivalence relation [a,b]~[c,d], if a*d=b*c. Real numbers can be constructed from rational numbers, too; there are two common ways: the Dedekind cuts, or the Cauchy sequences.
I have figure out something like this myself (probably it misses a lot of things, I just played around some ideas) : I started philosophically saying that there are 2 things: "Something", and "nothing". "Nothing" can mean zero. I make a box, it is "something". If I open it, it contains something -which is a box-, or nothing. Opening the box could be a "previous" function. Is this correct?
One of the things I really like about homotopy type theory is that it encodes this idea of "representation doesn't matter" into its equality operation; If you have ℕ defined as the Zermelo representation, and ℕ' defined as the von Neumann representation, one can prove ℕ = ℕ', and from there, ∀(P: Type → a) P(ℕ) = P(ℕ'); in other words, any construction on the Zermelo representation is propositionally equal to the construction on the von Neumann representation; any proof on any representation generalizes to any proof on a type that is propositionally equal to that representation. Thoughts?
ubsan You can do essentially the same thing without HTT; that's sort of what axioms are for. But I admit the relevant perspective is probably more natural when looking through an HTT lens.
diribigal I think the axioms mean you can write a proof that relies only on the axioms, and generalize it to all objects that satisfy those axioms; however, I don't know if, strictly, you can write a proof using the structure of the objects and translate to a different structure that satisfies the axioms.
If you have a hard time picturing something coming from nothing just substitute 0 for balance in your head. So in the suitcase example picture two sets being compared to one another on differing axioms. subtract one axiom from the other and if it comes to zero both axioms are balanced. if not, then one axiom compared to the total value of both has indeed created a suitcase unto itself out of nothing but in reality has taken a suitcase from the other axiom as compared to the whole of both axioms. If you have 10 suitcases and give 10 suitcases away you have zero suitcases but they didn't stop existing in terms of nothingness. They just now belong to someone else.
The separation of the idea of "numbers" from the idea of "cardinal" meaning, in my opinion, means that they aren't "numbers" at all. Numbers exist to be cardinal, at least when it comes to most applicable mathematics. Sure, a language or society could potentially have an ordinal system and a separate cardinal system, but the cardinal system is the one that I would call "numbers" whereas the ordinal system would not receive that name. Therefore, in my view of the purpose of numbers, the Peano Axioms do not define the natural "numbers" because it does not enforce cardinal interpretation.
I disagree. Most languages do have separate cardinal and ordinal numerals (very often the first few ordinal/cardinal numerals even mismatch linguistically: one-first, two-second;). They both obviously represent numbers in a very true sense. Peano Axioms capture the "numberific" essence that sets apart numerals from other quantifiers (like many, few, some, etc.).
...The Later-Wittgenstinians, of course! :) They wouldn't be happy about the idea (suggested in the video title) that the natural numbers are "made of" or "composed of" or "reducible to" something more basic or fundamental; i.e. that there is a logical structure underlying it all to begin with. But, to be fair, in the end, the presenter does state the contingent nature of these logical ideas, so the Later-Wittgenstinians would probably just be grumbling a bit over the inconsistency of the various formulations :P
Jon Sebastian reductions can be helpful but they tend to introduce irrelevant features such as "3 € 4" as mentioned by other comments on this video. Natural numbers can be just what they have to, as mentioned last in reference to Dedekind. Live with the fact that the axioms only describe the numbers up to isomorphism.
It's weird to consider the empty set ( i.e., nothing?) as something. But, once we make that leap, everything seems to become possible! Is most (or all) set theory like this? That is, the study of the different structures of the empty set?
Except the empty set is not an empty suitcase, neither is it a "container". It's a mathematical object that represents nothing. That is, we define nothing to be something, and that something is called the empty set.
No, its not. The empty set is the set that contains nothing. It is not nothing. Just like a set that contains an orange is not an orange. Understand the difference between a set and its contents is basic set theory.
A set is defined by its content alone. Those { } we use is not part of the set... It's just a notation. For example, a set containing exactly the numbers 1, 2, 3 is the same set as another set containing exactly the numbers 1, 2, 3. Hence, a set is not a container, rather, it is a collection of object(s). As such, a set with no object in it, i.e. with nothing in it, does represent the concept of nothing, since after all, a set is defined by its content, and this one happens to not contain anything.
A set is defined by its contents but it is not the same thing as its contents. If it were then set theory wouldn't work. If a set IS exactly the same thing as its contents then the set that contains the empty set is also the empty set. Which immediately makes all of set theory impossible. All sets are mathematical objects, including the empty set, they are by definition not "nothing". The importance of the empty set is that it is the simplest thing we use in set theory, a collection with no members.
Excellent lecture. My only objection, with all my due respect, is the back side music. It is disturbing, annoying and has negative effect on a better comprehension. The computer technology is advance enough to make the back side music optional for those who may like to have it. In a live class where you are presenting lectures do you also play music to go with it?
well, the peano axioms demonstrate that numbers are not really specific entities. They are more of a property of things within some system, be it sets, apples, money, bits, energy,... as long as the system satisfies peano axioms, the things within that systems are numbers.
what I thought. I get the feeling he went on a whole rollercoaster ride of critical thinking without regarding the simple fact that the concept of number is a reference to/by-product of naturally occurring quantities found throughout the universe
Well, he does bring up a point at 2:04 that mathematicians of the XIX and XX centuries were very uncomfortable with this vagueness, and wanted to put all of numberhood on firm, solid footing.
So in terms of programming concepts, the Peano axioms are the interface, and Zermelo's and von Neumann's models are implementations.
I would say that's a good analogy.
Great analogy
The axioms go a bit beyond what is usually considered an interface in say Java.
In Java an interface for defining the class "WholeNumber" would declare a defined constant "ZERO" and a function "WholeNumber successor(WholeNumber )", But there is no way to describe any constraints on how successor() and ZERO behave.
For example, you can't say in a Java interface that ZERO cannot be the output of successor(x) no matter what x is.
what if you were to introduce the concept of type traits and constraints (thinking meta c++ here) into the mix to define what (types) the results of S (and x (and ZERO)) are - the type manifests the briefcase/doll concept, so each different type in the system represents a different number..
I think you are overthinking; it's just an analogy :)
It's pretty clear that programming languages have problems (by design) when representing structures without actual concerns about computability and resource handling, and Mathematics is purely structure.
As a side note, I know that some cathegory theorists use an "extended Haskell language" sometimes as an alternative to abstract algebra and arrows diagrams; it's a cool language to express mathematical ideas algebraically.
In the span of 7 hours almost all of my favourite channels uploaded a new video. Today was good.
Agree!
and now 3blu1brown did as well. Perfect
Raphael Schmidpeter and twice in the same week!
This could be my favourite episode so far.
Great topic, interesting, deep and even philosophical, but set up in a way that makes it accessible to pretty much everyone with an interest in this kind of thing. Great job!
You should definitely make more of these - the really technical stuff, although interesting, isn't always accessible to many of us still busy with our undergraduate degrees : (
This was my favorite video in this series. Thank you for making it.
Agreed.
The set theory stuff comparing Zermelo and Von Neumann is really cool. Thank you PBS Digital Studios for all that you do.
He said "I'm sure a lot of you will disagree", but I agree with his answers to those philosophical questions.
4:28 So numbers are like ogres?
I am very glad PBS, the BBC, and others put educational videos on UA-cam.
Might as well do surreal numbers next! They get way too little attention!
At least they aren't imaginary. 🙄😉
It's going to be a bit hard to introduce them without digging into games first. Would be sad to gloss over why they work and what they "mean".
What about HYPERREALS? Or P-Adics? Or supernaturals?! I know WORDSSSSS
Those aren't Real Numbers!!! lol (Sorry had too)
Yes please!
I like the video, and I really respect that you discuss comment points and make clarifications at the end. Good engagement with your audience I'd say.
If someone wants to understand 9:27, you can try to prove the following: Every countable set can be equipped with a function S so that it satisfies the peano axioms.
Even more exact: If you have a bijective function f from the natural numbers to a set X, then there is exactly one function S' on X so that S' composed with f is the same as f composed with S, and S' turns X into "a set of natural numbers" (fullfills the peano axioms)
Nitpick: every non-finite countable set.
This is more or less exactly the definition used in category theory, namely that natural numbers is universal with that property.
And in the set theory ECTS we says such a set exists, full stop. No need to construct them from any some "primordial" soup like von Neumann/Zermelo constructions
I learned "countable" with the definition "having the same cardinality as the natural numbers". but yes, infinite countable set
So interesting! This is the first time I've heard a lot of these concepts, but I'm always impressed at how well you explain them. What a fascinating way to think about numbers! Thank you!!
Gabe, I love the personal wisdom you provide at the end of the video.
Regarding which model is preferred, I think the VN model is beautiful for its ability to convey arithmetic in simple set operations. The less than as a subset you mentioned already. Addition in VN is the size of the disjoint union. But of course we have to be careful saying size so we get around that by saying that we want the "pure set" that is isomorphic to the disjoint union one that is all fuddled with duplicates.
Thanks for another wonderful video Gabe. Have a great weekend!
When algebraic structures are generalized they become more powerful and you can find more mathematical objects that behave like the original structure you abstracted. Examples: vector spaces, groups, rings, fields...
So, when the structure of natural numbers is abstracted, is natural that somente will find other things that behave like them. That's awesome and beautiful.
"What are numbers made of? [...] Nothing, provided at least that you stipulate nothing exists."
Incredible statement.
The object of mathematics is not numbers but mathematical language. Or, to state it in linguistics terms, math is a metalanguage that operates on its own concepts (containers). It so happens that the containers/concepts of math are paradoxically the 'content' of Nature. So mathematical language seeks to be the projection (or subsumption) of Nature into an intensional set of concepts.
I feel like PBS is an undying teacher to me... First the alphabet with Sesame St. when I was 5, and now undergraduate math 15 years later.
I've had a very interesting convo with a prof.
He suggested that on any infinite set X (so there are injective functions onto itself that are not surjective, take some such F). You take an element of the set outside the image of F, and take the chain of that element.
It's easy to show that the set we get from the chain (name it C) and some operation "+" -- recursively defined by F restricted to C -- form a commutative monoid isomorphic to ℕ.
Interestingly enough, you can get some ordinal structure by finding other elements outside of F's image and "glueing" those chains together. You get a weird order where every element outside of F's image is made into a ''limit ordinal''.
I'll try to put up a reply video in the next few days, really liked this video and the way the alternative constructions were explained.
Second! = {First!}
308th={1,2,3,4,5,6,7,...,306,307}
@Gabe, PDF contains error. emailed ya. "30214/143" is missing the operator "+" between "302" and "14", to provide the correct interval between valid clock configurations. alternately, change "30214" (the incorrect number of seconds in a day) to "43200", and this is also a valid fix. will keep an eye out for anything else.
Awesome. Lemme fix that. Thanks!
+PBS Infinite Series
Since you've done videos on sets, foundations, functors, topology, and logic, would you ever consider doing a video on Topos theory?
Sure. Category theory is Tai-Danae's specialty, so I'd say there's a high likelihood you'll see something along these lines eventually.
Really like this episode and the previous one. Thank you for making these videos! My personal view aligns with yours. It is meaningless to ask the meaning of anything in mathematics, all these objects emerges one way or another from some fundamentally irreducible assumptions.
The suitcase concept reminds me of blockchain, in the sense that each new block is built on the hash of all previous blocks. The lesson would even synergize with the videos on cryptography and hashing.
Edit** PBS please do a video on blockchain!!
Once again brilliant work!
Nice! Von Neumann!!! Excellent episode! Thank you again!
I suppose it's fitting that in set theory, everything - starting with numbers - is made from (you guessed it) sets. I like that fact that the empty set is 0, then the next set (1) contains 0... just like computers start counting at 0, rather than one. I prefer the Reese's Peanut Butter Cup Theory of Everything, where 0 is an empty wrapper... tavi.
What is that Electronic Intro Music starting at 0:17? I am kinda obsessed with it, its so minimalistic but rich. Is there more of it somewhere?
The dreams that stuff is made of. The thing of shapes to come.
If 3 is really the same as {0; 1; 2} it would mean that you should be able to iterate over numbers in a for loop in programming!
In Python, if you want to iterate over 3, you'll have to write "for i in range(3)". In any other language, you'll have to write "for(i=0;i
Great video! I think it would have been nice to put more emphasis in distinguishing in the beginning ordinal numbers and cardinal numbers. While watching the video I kept thinking to myself that the numbers you define do not "feel" like numbers because I was thinking about cardinals.
That's coming up in the next few weeks. --Gabe
Great episodes, love this fundamental stuff! I have a request for an episode on Grassmann numbers, the exterior algebra, the Berezin integral etc. Maybe in cooperation with PBS spacetime on supersymmetry? :) Please :) Thanks! :)
at 9:50 you mention there being multiple ways to construct R out of N. I've only ever seen the Dedekind cut version presented previously on this channel. I'd like to see an episode with more detail on some of those ways.
J Halson try to look up Cauchy's constructions
Or Conway's Surreals
I see a distinct rise in quality of this channel's content. Nice concepts covered with little nuance (and if there is it's mentioned for people to go look up). Keep up the good work.
Very nice explanations, the video was very clear. Nice one. team.
This was exactly the explanation I needed
Well, the interesting question is asked in the second half of the video - anyone can see upon further reflection that these set-theoretic constructions formally parallel operations with numbers when the right relations are specified, but this doesn't of itself show that what a number is is a set. It especially does not show that 1-as-such is an operation in which 'nothing' is taken to be an object of the intellect (which seems to be Cantor's conception of what a set is). At least intuitively, this is not what what we mean when we say that there is _one_ thing - we do not take the empty set (0 in this formalism) to be an object of the intellect and conjoin that with the quantity of that thing. In a similar way, when I say there are two things, I do not mean "take now as an object of the intellect both 'nothing' (0) and another object of the intellect which is in turn 'nothing' (1)". This set-theoretic construction has always seemed to me to be formalism (a position in philosophy of mathematics) gone wild and not saying more essential to number-as-such - I think the Peano axioms do tell us something, but not set-theoretic constructions like these.
But, when it comes to math, I'm a filthy Platonist (realist), so my solution for now is to say that the essence of number is that which is in common, the form, of all these constructions, with individual constructions being different names of the same form; this is what I take to be the insight that these Peano systems are all isomorphic to one another and what the general importance of proving two things to be isomorphic in math is.
If you expected that the construction would tell you anything more about the Peano Axioms or Peano Arithmetic, of course it won't. It is a construction, a demonstration that the set N used in the Peano Axioms can in fact be described (if you're a Platonist you should consider that important). Some mathematicians are constructivists and demand a realization of an argument before accepting it.
Hi, thanks for these videos, they were great. But I have some questions:
1)How we define "function" when we construct natural numbers using Peano axioms, for example, can the notion of function itself be constructed in terms of sets ?
2)You sad that other formulation of the Von Neumann model is "A natural number is a set X, each element of which is also a subset of X". Maybe I misunderstood something, but don't any set satisfy this condition ?
3)And last, in the first video you sad that "=" is any relation that is reflexive, symmetric and transitive. But then what means for example that function of some input equals to some output, in the case of the video, what means S(Zelda)=x ? (I suppose that this question is closely connected to my first question).
Thanks in advance. And thanks for the videos.
Great questions!
1) Yes, a function can be constructed just through sets, but it's a bit tricky. First, we need to define the notion of a "tuple", meaning an ordered pair of _things_ (not saying "numbers" here on purpose, as it could be anything). If you have the tuple (A,B), how would you define that as a set? Well, you could do {A, B}, but that loses the order, as that would also be the representation of (B,A). Instead, the definition of a tuple is {A, {A,B}}. The representation of (B,A) would be {B, {A,B}} which is different.
Now, you can ask yourself: What even is a "function"? Well, it's something that maps objects from one set S to objects from a set T. But every mapping from one thing in S to another in T is just a tuple (A,B) where A is in S, and B is in T. So, a function is just a set of tuples {(A1,B1), (A2,B2), (A3, B3),...} where the A's are in S, and the B's are in T. Additionally, all the A's need to be distinct, as you can't map the same A to different B's.
If the tuple (X,Y) is an _element_ in the function f (because, as stated, f is just a set of tuples), then we write f(X)=Y. What that equals-sign means is answered in 3).
2) Not quite. For example, take the set "{Dog, Cat, Mouse}". The elements of it are "Dog", "Cat", and "Mouse". However, none of these elements are a set at all, so they're not subsets of the original set. "{Dog}" is a subset of "{Dog, Cat, Mouse}", but that's not the same as "Dog". Even if "Dog" is a set, it is certainly different from the set "{Dog}" which contains just one element. It's the difference between having a dog with you, and carrying that dog in a suitcase.
In the VN model, the number 4, for example, is just {0,1,2,3}. However, the number 2 is just {0,1} which is a subset of 4.
3) He glossed over what he meant by "relation", but it's also just another set of tuples, except with even less restrictions than functions. In fact, a function is just a specific _type_ of relation.
The set {(A,B),(A,C),(B,C)} is an entirely legitimate relation, but not a function because A is "mapped" to both B and C. In this case, it's transitive because (A,B) is in it (meaning A = B), (B,C) is in it (meaning B=C), and (A,C) is in it (meaning A=C).
It is neither reflexive nor symmetric, though. For a relation to be reflexive, it must contain the tuple (X,X) for any element X. For a relation to be symmetric, it must contain (Y,X) if it contains (X,Y) for any elements X,Y.
I mean it when I call those great questions, by the way. Because it takes a keen understanding of mathematical thinking to notice that Gabe left some things undefined, which at first glance seem obvious. That's the sort of thinking which got us to the Peano axioms in the first place.
Yes functions can be created using only set theory and a system of logic. A function is simply a set of ordered pair where the first element is the "input" and the second element is the "output" (actually this is a more general concept called a relation, you need to add some restricts to get a function specifically). And yes ordered pairs can be defined in set theory too but doing so is both incredibly difficult and largely pointless.
Watch this series. It will explain everything: ua-cam.com/video/CMWFmjlB8v0/v-deo.html
1) Yes it can. A function F from set A to B is a subset of AxB. So every element of F is a tuple (x,y) with x∈A and y∈B. That makes F a so called "relation". In order to make F a function, for every x∈A there has to exist a y∈B such that (x,y)∈F. Furthermore that y has to be unique.
There you go. A function from A={1,2} to B={a,b,c} could look like this F={(1,a) , (2,c)} and you can change to the common notation F(1):=a, F(2):=c.
F={(1,a) , (1,b) , (2,c)} would not be a function, because the image of 1 is not unique. You would say "F is not well defined".
2) No, that is not true for every set. Take this set A={{x,y},z}. {x,y} is an *element* of A, but *not* a subset. In order for {x,y} to be a subset of A, it would require that x and y are elements of A, but A has only two elements namely {x,y} and z.
3) Like in the first point a (binary) relation R is defined as a subset of a cross product of two sets A and B (in the case of an equivalence relation, A and B have to be the same set). So R⊆AxB. What we commonly mean by "=" is a so called equivalence relation which is again just a relation with certain properties, namely
reflexive: If x∈A then (x,x)∈R
symmetric: If (x,y)∈R then (y,x)∈R
transitive: If (x,y),(y,z)∈R then (x,z)∈R
S(Z)=x is just another notation to essentially say that the tuple (S(Z), x) is element of that equivalence relation we call "=", where S(Z) is the unique element y such that (Z,y)∈S.
edit: I notice that I just assumed it is okay to use tuples, when we only have sets. However tuples which are nothing more than ordered sets can be constructed by using unordered sets. One way to do it is (x,y) to be defined as {x,{x,y}} and voila, if I give you the set {{3,2},2} you immediately know that I mean the tuple (2,3) and not (3,2).
All these constructions using only sets can get really confusing, but it is also funny to think that you can rewrite something like 2=1+1 as {2,{2,+(1,1)}}∈= and even more confusing as as {{{},{{}}},{{{},{{}}},+({{}},{{}})}}∈=
In every academic setting in which I have been, if "function" is being formally defined, it has been defined as Treufuß did. Nevertheless, I confused an entire classroom of my fellow students when I described a function between two small sets by listing the tuples.
Very good to follow presentaion stile! Visual aids are a godsent;-)
You mentioned how you can construct the projective plane as a quotient space of a square. The frustrating thing about this construction is that it is not smooth at the point corresponding to the corners of the square, unlike the torus and Klein bottle constructions. If the Pacman board was a projective plane instead of a torus, and Pacman runs directly into a corner, the direction in which he leaves the corner is indeterminate.
By the way, the projective place is probably my favorite space.
"That's cute."
It somehow feels that boolean values would be a good stepping stone between natural numbers and nothing. Do they have any part in the world of number sets?
At 7:26 he says about the Von Neumann construction: "Every number is a set each of whose elements is also a subset of the set". This is true for the Zermelo construction, right? Except that Zermelo only as the "predecessor" element as a subset, but Von Neumann has all predecessors as elements. I am wondering if I am missing a deeper significance of elements being subsets itself ... the neat thing is just that all predecessors are subsets?
No, it's the von Neumann construction. Every element of an element of a vN ordinal is also an element of that ordinal. For example, 3 = {0,1,2}, with 0 = {}, 1 = {0}, and 2 = {0,1} all being subsets of 3. If we take 3 = {2} = {{1}} = {{{}}}, then 2 is not a subset of 3.
Numbers for poly-semiotic observation by any neural network are made of quasi crystalline-- E8 recurrent-- time crystalline glyphs for neurAl network use and interpolation into ideAl symbols then iconic signifiers then axiomatic stabilizers (of toric code as seen in quantum computers).
Does that mean the way we note number is a kind if logarithmic compression where the base are a rule for progressive nested set ?
That foundations of mathematics WAS developed in order to reduce the number of initial assumptions, that's not really a matter of debate. It's a historical fact. Higher mathematics encountered a bunch of problems caused by faulty assumptions. Mathematicians wanted to avoid that kind of thing.
Alexander F that is true, though Zermelo had a very different conception of what a set is than Cantor had, which is somewhat fascinating from a historical PoV. Check out the research article by F.W Lawvere on Cantors concept of Lauter Einsen.
However, modern interpretations do not in any way have to follow historical motivations (modern physics is plagued by this phenomenon). So there is still room for debate if one wants to, depending on how the question is phrased.
Re: "It doesn't matter what things in math really are", I certainly agree. Most of what we do in math relies on objects' properties rather than what they're "made of". Both of the constructions shown are no more than a few centuries old, but we've been using the natural numbers and proving things about them for millennia! The ancient Greeks used geometric constructions, but the exact same results with the exact same proofs still hold today.
I'd even go so far as to say the properties are more fundamental, since we choose the constructions so that we have the properties we want. The Peano axioms were chosen so that they satisfied the intuition we already had about ℕ, and the set constructions were chosen to satisfy those. Not the other way around.
To put it programming terms, we use the interface and only have the implementation for the sake of having one.
Your philosophy of numbers blows the foundation away of the French philosopher's Alain Badiou 'ontology'.
I get the feeling he went on a whole rollercoaster ride of critical thinking without regarding the simple fact that the concept of number is a reference to/by-product of naturally occurring quantities found throughout the universe
If zero is an empty collection. Then a more general question would be: what is memory? Memory is only an association between two "things", so it could be represented as a network. Numbers are then a special case of a network where all associations are sequential, having a beginning and an end because otherwise it would be a circle..
I would say that numbers as physical properties are made out of physical objects.
But numbers as abstract objects are made out of constructions and axioms.
Please do a video explaining how to "derive" the integers, rational and irrational numbers from the natural numbers.
Something just popped into my mind while watching this and it was very interesting. Can you have an axiomatic system such that its axioms contradict one another? It might be a useless system but that's not the point? Or am I just being stupid?
Yes, in fact it is not known if Peano's Axioms define such a self-contradicting system.
I ca make a tuple theoretic model for the finite surreal numbers. That would be the Dyadics, which are the numbers you find on an old fashioned ruler with imperial divisions (fractions of inches).
For the 3 handed clock (hours, minutes, seconds) "swappable times" you'd be naturally limited to those times when one of the hands bisects the angle between the other two. Which hand that is in the middle will vary. I've no idea if there are any valid examples of this.
7:23 I didn't get it. Can someone explain?
me too. I typed the same question, and then came across yours. Let me know if you figured it out.
That's the von Neumann definition of natural numbers: any natural number is realized as a set of all numbers less than itself. So 0 is an empty set, 1 is {0} (set whose only element is an empty set), 2 is {0,1}, 3 is {0,1,2}, and so on. Observe that any natural number n is an n-element set. In addition, observe that natural number n+1 is obtained from n by adding n itself to it as a new element - n+1=n∪{n}.
Under this definition, natural numbers are ordered by the ∈ relation; we can define a
Interestingly you had to define a function. I think the best way of defining natural numbers is in terms of functions themselves, like in lambda calculus. Please, make videos on category theory.
¥δΣΩφ You can define functions from sets only using ZF.
¥δΣΩφ also Homotopy type theory
Tai-Dinae is a category theorist. I'm sure she'll do a category theory video if they let her.
Alexander F don't you mean a realm-theory episode ;)
She already invoked category theory with the episode which used the fundamental group
You can define sets from functions using only ETCS
Choosing between the Zermelo and Von Nuemman reminds me of the Axiom Choice.
Is mathematics formalized in a discrete manner then? Is it possible to think about the foundations of mathematics in a continuous fashion instead?
Regarding 9:07, doesn't that statement contradict the existence of non-standard models of Peano arithmetic? (Of which there are plenty by the Löwenheim-Skolem theorem.) I don't see how the standard model could be isomorphic to a non-standard one.
Löwenheim-Skolem is about 1st-order logic and would apply to a 1st-order formulation of the Peano axioms (and models thereof). But the version of the axiom of induction that I reference in this episode & in the previous episode "What Does It Mean to be a Number" is a 2nd-order statement. This is precisely the thing I said I didn't want to get into in my disclaimers at 3:21 (second bulleted item there), but the question you raise is valid. Short answer -- it's a 1st-vs-2nd-
order issue, and a 2nd-order formulation of the Peano axioms is categorical (all models thereof are isomorphic).
@@pbsinfiniteseries But then you're going to make the set: "{0, 1, 2, ...} + something". Sure, second-order Peano arithmetic has an axiom: "given any subset of natural numbers, if 0 is in the set, and for any n it also contains S(n), then the set contains all natural numbers". So this set would seemingly not comply with the axiom, because you could take the set {0, 1, 2, ...}, which indeed contains 0 and for any element n it contains S(n), but this set does not contain all elements. But here's the problem: who said that {0, 1, 2, ...} were a set? (Second-order PA only has a unique model up to isomorphism within a particular model of set theory.)
This one is on the point!
Numbers express properties, qualities, values, and flavors as expressions of consciousness.
I haven't watch this yet, or the previous episode. But I will with interest (even though it kind of goes over my head). I'm wondering if "cosmic black holes" are related? - I watched an introductory lecture by Leonard Susskind on black holes (I actually learnt stuff - the key things he wanted people to take away). One of things I took away from it (and I expect I have this wrong) is that it may well be that there is nothing inside a black hole? - not because the stuff inside the black hole "vanishes", but because of the breaking of entanglement at the "fundamental" level of information, i.e. like an entangled virtual partice pair forming near the event horizon of the black hole and splitting, for one particle to fall past the event horizon towards centre of the black hole and the other to escape, and in the process a photon is created that also leaves the black hole. The idea is the escaping particle loses entanglement with the particle inside the black hole, but entangles with the escaping photon instead. So information is conserved about the particle in the black hole - and by virtue of this, it doesn't need to exist and so, some how, it doesn't? (this is why Hawking Radiation causes black holes to loose mass?). Though at the same time the particle falling into the balck hole could exist as scrambled information on a holographic encrypted event horizon - though whether this consitutes "existing" if it is "fubar-ed" I don't know? I'm wondering if the same kind of vanishing trick applies to number sets on a fundamental level? - that you can never reach a "singularity" or "zero", but yet these concepts still logically and physically seem to still exist by virtue of what is around them. Like with the arrow of time, it seems like such things are probabilistic? Bip. Thanks.
Chris: So-called (spinless) black holes are just Schwarzschild solutions. They don't have an actual singularity and can be analytically extended well inside the Schwarzschild radius. A lot of misunderstanding comes from treating the distant, nearly flat background as applying "within" the black hole, whereas spacetime becomes extremely curved as you get "inside" it. If proper care is taken to identify the time direction, you can create an Einstein-Rosen bridge, aka wormhole, everywhere smoothly differentiable.
Doug Gwyn Thanks. I neglected to mention anything about general relativity in my comment didn't I :P ...I understand relativity pretty well - like a layperson's understanding.
Zermelo is also a very delicious candy bar only found in eastern europe. *Mostly Nougat.*
The minimal construct of everything organized must be broken down to elements of thinking consistently. That just made me think, is there a minimal set of axioms that define a set of unintelligible sets? Maybe its a stupid question in that its only those things which are opposite of any logical construct. Like for example, any number in a set is not a set of itself.
I'm no mathematician, but I love asking stupid questions :).
To get incoherent sets you don't need any axioms. Assume anything you can describe is a set. Now is there a set which is "the set that contains all sets"? There should be since we can describe it but then we run into paradoxes.
what's your favorite construction of the reals? :)
A field R with the leat upper bound property, that coantains Q as a subfield.
Michael Novak that's more like a definition (like the peano axioms) not a construction (like the von neumann construction). My favorite is probably one that involves equivalence classes of Cauchy sequences of rationals, but I also like the simplicity of the reals as finite surreals that are simply approximated by rationals.
There is a pretty nice one that is called the Surreal numbers by J Conway. It actually constructs a bit more than just the reals.
I don't have one. I don't like reals at all, because they involve an uncountably infinite amount of incalculable, unknowable numbers. I think it would be fascinating if there would be a commonly known set of numbers that have some nicer properties. Defining numbers in terms of Turing machines that output Cauchy sequences would be one, but even then there's all kinds of problems with the halting problem, which leads to problems with undecidable equality and ordering between numbers...
+Pyry Jontio
Agreed. There is no valid mathematical construction of the reals, especially the unconstructible numbers (a subset of the reals). You can get away with constructible numbers I'll give you that, but even that requires an a priori notion of a function which is a very vague object for building foundations on (even more vague than sets). But when it comes to Infinite choice, Cantor's infinities, that kind of math, yeah nah it's all bogus.
You've been watching Wildberger haven't you :)
If one was to apply these axioms to the physical world, does it mean everything is made out of nested sets with a seed of nothing?
“numbers are made of candy” -clone phoebe
So the limit of the natural numbers moving towards infinity is the natural numbers itself, right?
It's like Kelsey never left .... referring to the awkward ramble session at the end here...
"Pacmanified Cube" is my new band name.
Assuming the Van Neumann construction is correct and 3 is really the same as {0; 1; 2}, what does a non-natural number look like? I am referring to negative numbers, rational numbers, and irrational numbers!
This is covered in the video "Crisis in the foundation of mathematics". Integers are a closure of integers under the operation of subtraction. So we define them as pairs of natural numbers [a,b] (by convention written as a-b), under the equivalence relation [a,b]~[c,d], if a+d=b+c.
Rational numbers are a closure of integers under the operation of division by a non-zero numbers. So once again we define them as pairs of integers [a,b] (by convention written as a/b), under equivalence relation [a,b]~[c,d], if a*d=b*c.
Real numbers can be constructed from rational numbers, too; there are two common ways: the Dedekind cuts, or the Cauchy sequences.
I feel like I can finally wrap my mind around Church numerals now.
At 6:24 doesn't the output suitcase contain 4 elements and not 2 elements?
Correct, at 6:24, the output suitcase Y contains 4 elements: a, b, c, and {a,b,c}.
I have figure out something like this myself (probably it misses a lot of things, I just played around some ideas) : I started philosophically saying that there are 2 things: "Something", and "nothing". "Nothing" can mean zero. I make a box, it is "something". If I open it, it contains something -which is a box-, or nothing. Opening the box could be a "previous" function. Is this correct?
One of the things I really like about homotopy type theory is that it encodes this idea of "representation doesn't matter" into its equality operation; If you have ℕ defined as the Zermelo representation, and ℕ' defined as the von Neumann representation, one can prove ℕ = ℕ', and from there, ∀(P: Type → a) P(ℕ) = P(ℕ'); in other words, any construction on the Zermelo representation is propositionally equal to the construction on the von Neumann representation; any proof on any representation generalizes to any proof on a type that is propositionally equal to that representation. Thoughts?
ubsan You can do essentially the same thing without HTT; that's sort of what axioms are for. But I admit the relevant perspective is probably more natural when looking through an HTT lens.
diribigal I think the axioms mean you can write a proof that relies only on the axioms, and generalize it to all objects that satisfy those axioms; however, I don't know if, strictly, you can write a proof using the structure of the objects and translate to a different structure that satisfies the axioms.
Bollocks..you have already used numbers in defining a series of steps
are there any books on this?
This guy is great on Impractical Jokers
Am I the only one who consider the concept of number to be much simpler than the concepts like set or function?
If you have a hard time picturing something coming from nothing just substitute 0 for balance in your head. So in the suitcase example picture two sets being compared to one another on differing axioms. subtract one axiom from the other and if it comes to zero both axioms are balanced. if not, then one axiom compared to the total value of both has indeed created a suitcase unto itself out of nothing but in reality has taken a suitcase from the other axiom as compared to the whole of both axioms. If you have 10 suitcases and give 10 suitcases away you have zero suitcases but they didn't stop existing in terms of nothingness. They just now belong to someone else.
Asking what anything in blockchain "really" is... Is beside the point. At the end is just fancy math :)
The Zermelo construction? No one uses this construction, because going to the ordinals is impossible.
Pacmanified cube, not bad. I'd like someone to expand on this :-)
Best channel ever
Great job!
'Dollify' should be voted word of the year 😁
Just call isomorphisms an equivalence class and there's no debate anymore!
Absolutely brilliant video. Thank you.
Make videos on first and second order logics please.
The separation of the idea of "numbers" from the idea of "cardinal" meaning, in my opinion, means that they aren't "numbers" at all. Numbers exist to be cardinal, at least when it comes to most applicable mathematics.
Sure, a language or society could potentially have an ordinal system and a separate cardinal system, but the cardinal system is the one that I would call "numbers" whereas the ordinal system would not receive that name.
Therefore, in my view of the purpose of numbers, the Peano Axioms do not define the natural "numbers" because it does not enforce cardinal interpretation.
I disagree. Most languages do have separate cardinal and ordinal numerals (very often the first few ordinal/cardinal numerals even mismatch linguistically: one-first, two-second;). They both obviously represent numbers in a very true sense. Peano Axioms capture the "numberific" essence that sets apart numerals from other quantifiers (like many, few, some, etc.).
Yes but what I'm saying is that, in my view of numbers, "first" isn't a number. "One" is a number.
Is "last" an ordinal number?
"Last" is an ordinal, but it isn't a number.
The Wittgenstinians aren't gonna be happy...
Jon Sebastian in what way exactly?
...The Later-Wittgenstinians, of course! :)
They wouldn't be happy about the idea (suggested in the video title) that the natural numbers are "made of" or "composed of" or "reducible to" something more basic or fundamental; i.e. that there is a logical structure underlying it all to begin with. But, to be fair, in the end, the presenter does state the contingent nature of these logical ideas, so the Later-Wittgenstinians would probably just be grumbling a bit over the inconsistency of the various formulations :P
Jon Sebastian reductions can be helpful but they tend to introduce irrelevant features such as "3 € 4" as mentioned by other comments on this video. Natural numbers can be just what they have to, as mentioned last in reference to Dedekind. Live with the fact that the axioms only describe the numbers up to isomorphism.
Is there a video on first and second order logic? Because every time I see it it confuses me
Van Neumann, the man who hid luggage
Now, infinite ordinals are made out of smaller ordinals
It's weird to consider the empty set ( i.e., nothing?) as something. But, once we make that leap, everything seems to become possible!
Is most (or all) set theory like this? That is, the study of the different structures of the empty set?
The empty set is not nothing, it is a container with nothing in it. If I have an empty suitcase I don't have nothing, I have a suitcase.
Except the empty set is not an empty suitcase, neither is it a "container". It's a mathematical object that represents nothing. That is, we define nothing to be something, and that something is called the empty set.
No, its not. The empty set is the set that contains nothing. It is not nothing. Just like a set that contains an orange is not an orange. Understand the difference between a set and its contents is basic set theory.
A set is defined by its content alone. Those { } we use is not part of the set... It's just a notation. For example, a set containing exactly the numbers 1, 2, 3 is the same set as another set containing exactly the numbers 1, 2, 3. Hence, a set is not a container, rather, it is a collection of object(s). As such, a set with no object in it, i.e. with nothing in it, does represent the concept of nothing, since after all, a set is defined by its content, and this one happens to not contain anything.
A set is defined by its contents but it is not the same thing as its contents. If it were then set theory wouldn't work. If a set IS exactly the same thing as its contents then the set that contains the empty set is also the empty set. Which immediately makes all of set theory impossible. All sets are mathematical objects, including the empty set, they are by definition not "nothing". The importance of the empty set is that it is the simplest thing we use in set theory, a collection with no members.
Excellent lecture. My only objection, with all my due respect, is the back side music. It is disturbing, annoying and has negative effect on a better comprehension. The computer technology is advance enough to make the back side music optional for those who may like to have it. In a live class where you are presenting lectures do you also play music to go with it?
So far from my skills but so fascinating! :-)
i like how [] and {} nicely translates to python.