What are Numbers Made of? | Infinite Series
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In the physical world, many seemingly basic things turn out to be built from even more basic things. Molecules are made of atoms, atoms are made of protons, neutrons, and electrons. So what are numbers made of?
Check out the previous episode to find out What It Means to be a Number
• What Does It Mean to B...
And to see Gabe's solution to the Torus Clock Challenge check out:
bit.ly/pbs_clock_challenge
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Previous Episode:
What Does it Mean to Be A Number?
• What Does It Mean to B...
Torus Clock Challenge:
• Telling Time on a Toru...
How to Divide by "Zero"
• How to Divide by "Zero...
Blog post about the Peano axioms and construction of natural numbers by Robert Low:
robjlow.blogspot.co.uk/2018/01...
Recommended by a viewer for connections to formulation of numbers in computer science:
softwarefoundations.cis.upenn...
Any set N and function S that meet these conditions will behave , respectively, like the natural numbers and the operation "next" or "successor". You can even define operations that fully mimic run-of-the-mill addition and multiplication in terms of any suitable S, regardless of the details of how S works . In this sense, the Peano axioms distill numberhood down to its bare essentials.
Written and Hosted by Gabe Perez-Giz
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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
4:28 So numbers are like ogres?
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!
He said "I'm sure a lot of you will disagree", but I agree with his answers to those philosophical questions.
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!
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 : (
The set theory stuff comparing Zermelo and Von Neumann is really cool. Thank you PBS Digital Studios for all that you do.
This was my favorite video in this series. Thank you for making it.
Agreed.
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!!
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.
"What are numbers made of? [...] Nothing, provided at least that you stipulate nothing exists."
Incredible statement.
I am very glad PBS, the BBC, and others put educational videos on CZcams.
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!!
Second! = {First!}
308th={1,2,3,4,5,6,7,...,306,307}
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!
Once again brilliant work!
"That's cute."
Nice! Von Neumann!!! Excellent episode! Thank you again!
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''.
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.
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! :)
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.
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
Very nice explanations, the video was very clear. Nice one. team.
+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.
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 good to follow presentaion stile! Visual aids are a godsent;-)
This was exactly the explanation I needed
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.
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.
Does that mean the way we note number is a kind if logarithmic compression where the base are a rule for progressive nested set ?
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.
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
This one is on the point!
Great job!
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
Please do a video explaining how to "derive" the integers, rational and irrational numbers from the natural numbers.
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.
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).
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.
Choosing between the Zermelo and Von Nuemman reminds me of the Axiom Choice.
@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!
Absolutely brilliant video. Thank you.
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?
So far from my skills but so fascinating! :-)
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).
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
Zermelo is also a very delicious candy bar only found in eastern europe. *Mostly Nougat.*
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.
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: czcams.com/video/CMWFmjlB8v0/video.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.
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.
Numbers express properties, qualities, values, and flavors as expressions of consciousness.
Your philosophy of numbers blows the foundation away of the French philosopher's Alain Badiou 'ontology'.
Is mathematics formalized in a discrete manner then? Is it possible to think about the foundations of mathematics in a continuous fashion instead?
Is there a video on first and second order logic? Because every time I see it it confuses me
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.
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..
Asking what anything in blockchain "really" is... Is beside the point. At the end is just fancy math :)
Make videos on first and second order logics please.
Pacmanified cube, not bad. I'd like someone to expand on this :-)
"Pacmanified Cube" is my new band name.
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.
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.
This guy is great on Impractical Jokers
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.
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
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?
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}.
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.
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
i like how [] and {} nicely translates to python.
I feel like I can finally wrap my mind around Church numerals now.
are there any books on this?
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.
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?
Ok if I had suit case N how many suitcases would I have in total?
'Dollify' should be voted word of the year 😁
ℕth!
What a natural comment!
Could you show, how surreal numbers are constructed?
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...
Hey Gabe, can you please please please make a cameo in Space Time? 😍
The Zermelo construction? No one uses this construction, because going to the ordinals is impossible.
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.
Best channel ever
I like Von Neumann construction because I can make more relationships whit "Sub-Nets" :)
How can we get from the naturals to the reals tho? Like I get how to get to the algebraics but the reals are really a whole other type of infinity which you can't just obtain from the type of infinity in the naturals. Isn't this a huge problem in mathematics?
* Integers can be obtained from natural numbers as a closure under the operation of subtraction. (Define a set of pairs of natural numbers [x,y], by convention written as "x-y"; this set, under the equivalence relation [x,y]~[v,w] if x+w=v+y, are the integers.)
* Rational numbers can be obtained from integers as a closure under the operation of division by numbers other than zero.
* Real numbers can be obtained from rational numbers using either Dedekind cuts, or Cauchy sequences. (I won't go into detail of the construction right now; you can look this up on Wikipedia.)
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.)
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.
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 :)
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.
Carinals and then a CH reboot ?
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.
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.
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.
Numbers are made with SpongeBob imagination power.
Just call isomorphisms an equivalence class and there's no debate anymore!