There's something incredibly satisfying about a math video that makes you pause it and think for *just the right amount of time* before you grasp what it's saying. Well done.

@jasondoe2596

6 жыл бұрын

Haha, yeah, I also found the pacing excellent :D

I remember in high school picking up a random math book with a chapter on the Peano axioms. I was really happy that evening, and know it is one of my fewer high school days memories. Quite cool. Now I'm listening to you guys :)!

@enlightedjedi

6 жыл бұрын

And actually I never knew about the 5th axiom, but I did at the time quite quickly came to adding and multiplying and later exponentiation :)!

@pbsinfiniteseries

6 жыл бұрын

I wonder if we picked up the same book back in the day...

@alexanderf8451

6 жыл бұрын

Yes, the idea that there are notions more basic than even counting is amazing!

@enlightedjedi

6 жыл бұрын

@PBS It was in Romanian so I doubt it, but the idea stays :)!

God I miss this channel. Sci show, CrashCourse and the pbs channels are the only channels I’ve continuously watched since high school. I found this one 3 years ago(after high school) and it died 2 years ago. Still come back to it now and again. I hope is comes back some day!

Trying to tell what numbers are, Ok... lets start with Axiom 1..2,3,4,5...

@talideon

6 жыл бұрын

The axioms have no order. They're just a convenience for referencing them.

@veggiet2009

6 жыл бұрын

Gerard Tan touche

@ocean_0602

6 жыл бұрын

It's just a labeling convention. I guess he could have used letters, ordinals ("first, second"...etc.), or maybe given each axiom a name.

@Deguiko

6 жыл бұрын

Gerard, Thats a good point. You could get rid of numbers by merging all axioms into a single axiom that says "(A1) and (A2) and ... and (A5)". Then you can say that this is THE axiom that defines natural numbers.

@cabra500

6 жыл бұрын

Yeah and the video data is contained in a binary system that uses 1's and 0's. Good luck next time PBS

This vid is pure gold. I've never had Peano's axioms explained so clearly before! Thank you so much, this helps a lot!!!

I have dyscalculia and although I have a much better understanding about math than from when I was a young child, these videos show me a different understanding of numbers that I was completely unaware of before.

All the comments arguing that 0 is not a natural number are very annoying to me. It reminds me a very pedantic student of mine who refused to believe that "either x or y" could be interpreted as an inclusive rather than exclusive "or". There seems to be an issue in math education where people assume definitions are somehow absolute truths rather than matters of preference which are stated before a proof.

@wilddogspam

6 жыл бұрын

AFastidiousCuber it seems that many people never lose the childish notion that what "makes sense" to them is an universal truth. From that they seem to believe that arguing intuitively is allowed in math or logic, which annoys me to no end.

@mzg147

6 жыл бұрын

Discussing definitions is a very important matter in science, life and mathematics. From a purely logical view it is true that definitions are nothing more than a abbieveration which can be created arbitrarly. But, totally arbitrary definitions hinder understanding. In order for our understanding to be clear, definitions must be intuitively justified.

@paulcasanova1909

6 жыл бұрын

But your counterexample are just words. People are arguing that because it’s a set. The natural numbers [1, infinity) is not the same as the whole numbers [0, infinity) by definition of a set. You can change the words of the sets to A and B respectively, but the underlying content does not change. And this matters a lot for things like groups, convergence of series, convergence of functions, yada yada yada. Its a small difference, sure. But there are some things that relies on starting from 1 instead of 0. Just as with computer science, it matters to start from 0 instead of 1.

@alexanderf8451

6 жыл бұрын

+mzg147 The name used to describe N is not substantive and distracts for what is relevant.

@rufusneumann9703

6 жыл бұрын

math.stackexchange.com/questions/283/is-0-a-natural-number ... If you are unhappy with that, define your own set equal to [0,∞) or (0,∞)

I knew about peano's axioms but i never heard of something even more foundamental than them. Really exited to see next episoded

@alexwang982

5 жыл бұрын

LetsTalkAboutMath Brains?

Gabe is the best. I like his way of explaining physical and mathematical concepts.

It's so good to see you back again, Gabe! Thanks! 😊

I have been waiting for this video for years!

I don't have words. Thank you! I am really grateful to you.

Really glad to know that those submissions for the Metallic Ratio Challenge are still being looked over; I was afraid my submission would get completely lost in the shadow of further episodes. This is such a great channel, please do keep it up :D

@pbsinfiniteseries

6 жыл бұрын

I'm definitely on those submissions. It's just hard to carve out enough unobstructed time to go through them all carefully given the rest of the production schedule. My cursory look suggests none of what's been submitted so far qualifies as a bona fide proof that there is no such ratio, but I don't want to be hasty in drawing that conclusion. --Gabe

That was awesome!!

This is some of the neatest stuff I've seen on this channel.

I had to take notes and really study this one for a while. Solid stuff.

That straight face with which he said proving association, commutation and the distribution law is "super fun" though.

This is just so logically beautiful! Inspiring! Thank you.

I'm looking forward very excited for the next episode

Peano and Dedekind are great! Thanks for showing!

Thank you for this video! I felt taken on a journey... explained very well :)

Sponsored by Nintendo? ;)

@pronounjow

6 жыл бұрын

PPAChao I wouldn't be surprised. Nintendo does want to get more people into making games. Just look at Labo and Super Mario Maker as examples.

@bunbunnbunnybun

6 жыл бұрын

How

@sriramaniyer9415

5 жыл бұрын

😂😂😂

best video on this topic... thank you!

THANKS for another great video!!

Excellent lecture. Thank you

This is now my favorite math video on youtube. This video defines numbers in terms of sets, but it's possible to not even bring those up explicitly. You can axiomatize the natural numbers without ever explicitly mentioning the set *N*. Rather than thinking of numbers as objects, you can think of them as words that can refer to whatever you like as long as the way they are used is consistent with the theorems that follow from the axioms. There can also be different axiomatizations that produce different theorems, especially when it comes to induction. You can also produce more limited kinds of axiomatizations for the sake of aiming at nice properties, like completeness and decidability. Which axioms you choose to study and use depends on what aspects of numbers are interesting for you. Just as there are alternative geometries, there are alternative ways to axiomatize the natural numbers which produce varying results.

Excellent , you are doing as fine on this one as you did on space time (still missing you )

Great job, Gabe!

Wonderful! Thank you!

Love love this !

When you think of the abstractness of the number 15, you're still thinking of something 'real'. Some things are real, not in the sense that you can touch it, but in the sense that it is perceivable. Numbers are one of the best ways of proving that there is more to reality, other than just what's in front of us.

Nice episode ^^ thanks and keep up :)

Really helps! thanks!

Excellent & fascinating episode! I knew most of that (even though I'm not a mathematician) and it was very satisfying seeing it laid out so cleanly. Now the *question* ; it seems clear that, according to the Peano axioms, the more "traditional" or "intuitive" definition of the natural numbers (how the ancients used them) would just be the set N - {Z} . Is that all there is to it, or I'm missing something subtle?

@mrpedrobraga

2 жыл бұрын

You could define this easily by trading "Z in N" for "S(Z) in N"

Enlightening Video Sir🙏🙏💥💥💥

My brain almost broke that time. I know I probably should've felt that way because other times but this time I almost broke because I almost understood all of it.( Weird how that works.) I'm glad you made this video. I've been wondering how to explain the concept of what it means for numbers to be numbers to a four year old without forcing them to recall the order of seemingly random vocalizations (won, too, three, for, phyv) and then trying to explain the reason for that order to them.

It's funny that the exam I would have done tomorrow, if it were not for the snow, is precisely about logic, peano arithmetic included, nice video as always! :)

Wow so well explained

Great job on the video, can't thank you enough for this awesome series! I have one question coming out of it though; why can we assume the existence and underlying properties of a "function" S(x) within Peano's Axioms? Wouldn't that act the same as using the term "next" given that we can assume what the word means?

I think this is the best video on KZread

@pbsinfiniteseries

6 жыл бұрын

While I appreciate the compliment, *this* is actually the best video on KZread: kzread.info/dash/bejne/ZqZr2aqGl5i_mso.html

lmao the creator of this is my dad’s grandpa’s uncle

I'm glad peano's axioms exist. I mean i guess it would be okay if the simplest way to talk about numbers was the natural numbers but it feels like a property of the universe or something like that.

Loving that you guys chose Fibonacci numbers for the Patron Levels on Patreon.

Brilliant!

Man. This is some crazy stuff

Gabe you are my favorite host. Such good speaking skills and an awesome explanation to top it off. I already know this stuff..I am guessing you will use the Von Nuemann hierarchy of nested empty sets to build the naturals next time. You will go far my friend, in math or in show biz - choose wisely :-)

@pbsinfiniteseries

6 жыл бұрын

Nice of you to say! Not sure that's true, but nice nonetheless. Thanks for the compliment. As for the VN hierarchy, shhhhhhh! Don't let the cat out of the bag just yet.

@ChurchOfThought

6 жыл бұрын

PBS Infinite Series I watch a lot of edutainment - PBS Spacetime, Mathologer, Minutephysics, 3Blue1Brown..etc. I really think you are the best explainer / speaker to grace KZread. I would say PBS but we both know Carl Sagan and Mr Rogers have you beat ;-) On a random and fun note, ever hear about the duties of Von Nuemann's assistants? Enjoy: shitpost.plover.com/m/math.jobs-you-dont-want.html and the original source: www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Lorch219-230.pdf

Hey I found the Space Time guy! Space Time still my favorite KZread show.

Gaaaaaaabe! We missed you so much from PBS Space Time. I thought the last Infinite Series video might have had you as a guest host but you're back! Is this going to continue for a while? I really like Kelsey too, too many good hosts!

@pbsinfiniteseries

6 жыл бұрын

Waddup! Kelsey went on to finish her PhD. Tai-Danae Bradley have taken over the channel as co-hosts. This was all announced here: kzread.info/dash/bejne/gHypxdKzkqecYJM.html

Excelent video

I would love it know how you would do this same thing for the real or complex numbers, or if it is even possible.

This looks very similar to Church Numerals. Please do a episode on Monads and how they relate to describing many types of computations like Input/Output, nondeterministic computations, parsing, etc...

@jasondoe2596

6 жыл бұрын

Oh, *yes* please!

In first grade my teacher told me a natural number is a counting result, this ins't very mathematical but it's good enough as an informal definition

We miss you over at Space Time.

Answer to the explaining task: Words for counting individual articles.

Believe it or not - i thought about this very problem 2 days ago. And from peanos axioms, the set 2,4,6,8,10... also seems to satisfy all axioms. There is a smallest element, and there is exactly one succesor for each element. The function S(n) is then obviously n+2. The same goes for the set 3,6,9,... and the set 4,8,12,... and infinitely many others. But those sets are clearly not the natural numbers. So whats going on here? Where the mistake? Why does it seem like peanos axioms describe infinitely many sets and not just natural numbers?

@pbsinfiniteseries

6 жыл бұрын

My gut response (without having thought this through very much, so I'm happy if other people chime in to correct me) is that one of two things is at play (or both things). (a) One thing we could say is that you *did* just define the regular old natural numbers; you've simply *labelled* them with the symbols '2', '4', '6', etc rather than '0', '1', '2', etc. Try defining arithmetic on your set, i.e. define functions in terms of S that behave like addition and multiplication, and see if things work as you expect. (b) The evens (or the multiples of 3 or 4 or whatever) can be put into isomorphic correspondence with the more familiar naturals, in which case you've simply uncovered a different realization of the set N. I mean all this in the sense of using numbers as *ordinals* , i.e. just for ordered indexing, which is really what the Peano axioms capture. If you want to talk about *cardinal* numbers for capturing the notion of numerical quantity, then we have to be careful about how we define cardinality, and if we were careful and followed your construction, I suspect we'd conclude either that (a) your labels are just labels, and that your '2' is really a 1 (or 0, depending on whether you included 0 in your construction or not), your '4' is really a 2, etc, or (b) that you've left out a whole bunch of cardinal numbers. But again, I haven't really thought through what I just said.

@mickyj300x

6 жыл бұрын

the rules of multiplication in peano arithmetic force s0 to be 1. if you just have s, or s and +, then your example works

@cabra500

6 жыл бұрын

You must remember that this axioms are prior to the natural numbers, they come before the numbers. So you can't really say that (2,4,6,8) satisfies the axioms and the S(n) function because to have (2,4,6) you had to "create" [1,2,3,4,5,6...] and for you to make these numbers you already defined your S(n) function as n+1 (remember that the natural numbers COME FROM the peano axioms, so you can't define a set of symbols to be the natural numbers and then use symbols inside of this set of symbols to create the natural numbers again.) So, you can only define S(n) = n+2 and your numbers to be (2,4,6...) if you consider that there is no 1,3,5... and this would mean that your set would still be the natural numbers (as it always will be if you follow the peano axioms) but with different symbols.

@giancarloantonucci1266

6 жыл бұрын

One can define a bijection between even and natural numbers. Hence, they have the same cardinality. This means that in practice you have just relabelled the natural numbers as {2,4,6,8,...}. The same holds with any other set of this kind.

@jasondoe2596

6 жыл бұрын

zhadoomzx, that's a _good_ question, and my own gut response is the same as PBS's ; that *your* 2, 4, 6, 8, ... are indeed the natural numbers, *labeled* differently. Think for a moment: could you shove the "missing" elements in there, based on those axioms? The answer is no. 1, 3, 5, ... are unreachable. You just labeled 1 as 2, 2 as 4, etc. and now your unit element for _addition_ is named 2 instead of 1. (necessary disclaimer: not a mathematician)

Wow great video

Welcome back

[06:22] ordinals but not cardinals to find subsequences within a larger... Can the first succession equal the two next, and the four next-next...

"What does it mean to be a number?" Just ask one of Charli Damelio's fans

This episode is already my favourites. Please don't be too Platonic in the followup LOL.

Imagine two rooms; call them A and B. Room A contains nothing, while room B contains everything imaginable. Most people would think B is the room to go for, but the mathematician prefers A because she can easily prove B cannot exist, whereas she can use A to establish the entirety of mathematics. I got this from I believe life of Fred.

Great video! Just want to mention that the Peano axioms, as stated in first order logic, do not allow to isolate the standard model IN of natural numbers. Indeed the induction axiom formally states that if some property P holds for 0 and whenever P(n) holds, P(S(n)) holds, then P(n) holds for each n. Consider the set T to be the union of a copy of the natural numbers followed by a copy of the rational times the integers (Q x Z), in such a way that no element of QxZ is a successor (S of something) of any element of the naturals. Call 0 the first element of the naturals. One cannot find any first order property which "separates" our standard model of natural numbers from the model T. As an exercise try to find a property which separates the standard model IN from the model obtained as the union of two copies of the natural numbers, one following the other.

@destroctiveblade843

6 жыл бұрын

Gianluca Basso I have a question since you seem to understand this stuff , the axiom that you mentioned (the axiom of induction) is actually a theorem to us and we proved it from the axiom " every non empty sub set of N has a smallest element" wich wasn't mentioned here , so I am a bit confused bout that

@gianlucabasso

6 жыл бұрын

Destroctive Blade This is a great question which goes to the heart of the difference between first and second order logic. The way you state your axiom (known as well ordering principle) involves quantifying over subsets in addition to quantifying over elements: for every subset A, if A not empty then there exists n in A such that n is smaller than all other elements of A. Allowing quantification over subsets seems great, but it comes with some drawbacks (no completeness theorem), so logicians usually stick with first order logic.

@destroctiveblade843

6 жыл бұрын

Gianluca Basso now the no compleatness theorem ( as I hear) works for any set of axioms that can ever exist , and so the problem should still exist no matter what right ? And again if we take the induction axiom in the way you proposed it , we would not ve able to solve the problem you talked about , so I still don't see why it would not be a benefit to change the axiom in the way that I mentioned , (ps: we actually did the same thing for R , we used the axiom of the upper bound to prove the rest )

@gianlucabasso

6 жыл бұрын

Destroctive Blade I know it will be confusing but the incompleteness theorem (which holds for sufficiently strong and recursive theories, as Peano Arithmetic is) is NOT the negation of the completeness theorem (which does holds for all first order theories, including Peano Arithmetic). You can check them on Wikipedia to have an idea.

@destroctiveblade843

6 жыл бұрын

Gianluca Basso I miss read no compleatness for incompleation , it turns out that you must be extra precise in the wording hhhhh but I will definitely look this up

A lot of your videos have gone private, I really enjoyed them. Please make them public again if possible.

You will be missed ❤️

if they had taught me this before I learned about numbers that would've been great

Dang I remember getting being asked to describe numbers without the word "number" in high school. Why did this video not come out 10 years ago?

A video on tesselations please.

I had a feeling recursion was coming into this, if only because the use of the axioms had a recursive flavor to them. MY COMPUTER SCIENCE DEGREE ISN'T WORTHLESS. YAY.

Another way of thinking about natural numbers is treat is as feature of finite sets. My idea is to define numbers as equivalence class of relation of existing bijection between finite sets. BTW to define "finite" we don't actually need numbers.

thanks

Will the next video be on ZFC?

Numbers are instances of complete concepts. As instances increase the symbol we use to quantify the amount of instances changes to reflect that. , = A , , = B , , , , , , , , , , = J Half a "," = .E If we are assuming modulo J.

This is what people should say when someone asks why 1+1=2, as opposed to saying that it 'just does'. It'll give them a more appreciative world view how the most fundamental things can be grounded in something more much deeper.

And that is why Set Theory is so important. Some people think it is useless. But, it is like saying that pistons are useless in an engine. The "minimal" parts have to be kept really "small" and as few as possible and carefully chosen to use them later to be able to create the foundations of mathematical objects that will become extremely complex. These "minimal" and "fundamental" parts are called axioms. Axioms require no proof by definition.

Alright let me settle the dispute about 0. Definitions differ for the natural numbers. Some include 0, others exclude it. Peano's original axiomatization excluded 0. In my personal opinion, it is better to include 0 because it makes the natural numbers a commutative semiring, as well as a lattice under divisibility. Also, it makes denoting the set {0, 1, 2, 3, ... } easier: *N* instead of *N* U {0}. The set {1, 2, 3, ... } already has a nice notation: *Z* with a superscripted +

@Icenri

6 жыл бұрын

But, but, what about my existential crisis of having 0 as the first number? As a programmer it proved very counter intuitive to start labeling all my arrays with 0. :''( Edit: Just kidding, it's most probably right to start with 0.

@paulbucci

6 жыл бұрын

Maybe a little tougher to learn when you start programming, but it makes a hell of a lot of sense when you think about indexing multidimensional arrays in 1D. Suppose my 2D array is of size m by n, but I want to store it in 1D with row-major order (i.e., stack rows). The 1D index of any element i,j of my 2D array is i*m + j*n.

@TheClonerx

6 жыл бұрын

In history, the firsts natural numbers doesn't contains 0.

The axioms do seem to suggest progressions or “nextness” but there isn’t any particular defined interval. Surely the natural numbers exhibit a fixed interval. There are many functions that build on itself but they can result in ever increasing, or decreasing, intervals in the outputs. Or am I getting ahead of things?

Hello, are there any books you would recommend me reading? The topic doesn't really matter. :)

holy fuck............. this guy traveled through space time and reached the realm of mathematics... missed him on pbs

I've read some definitions. 1. Each natural number is a collection of sets with the same cardinality. 2. 0 = {}, 1 = {0, {}}, 2 = {1, {}}, ... They seems to be more natural and less assumption.

How can I derive divisibility just by usind add - like multiplicity ?

Re the metallic ratio challenge. What about polyhedra instead of polygons? And polyhedra in higher dimensions?

Do these axioms in any way define or find prime numbers?

Beautiful video. The explanation was very clear and sound. Now I'm just wondering: if Successor is a function, i.e. an arrow, doesn't it imply the existence of an input AND an output? In other words, my concern is that we assumed the distinct existence of such concepts in order to build the natural numbers. But I think that merely accepting a conceptual difference between "input" and "output", or between "this" and "that", or "a thing" and "another thing", however you want to call it, implicitly entails the notion of "one" and "two". 1 and 2 could just well be the names and the symbols that we assign to these concepts. In other words I believe the notion of "function" needs these concepts in order to exist, and therefore these concepts need to be more fundamental than it. Maybe 1 and 2 should be fundamental, and everything else should follow from them. (For example, after requiring by axiom the existence of 1 and the existence of something _other than_ 1, i.e. 2, I could define 3 to be something other than 1 _and_ 2, i.e. the "two" _of_ {1, 2}) What do you think?

So the set of natural numbers is just one possible set that satisfies those axioms right? Why do we choose the set we did and not another one? (For example a set with -3.65 as first element?)

as a lowly computer scientist I think of functions as number operators so I wouldn't have thought of this!

The objective here seems to have been to separate natural numbers from the idea of counting (finding a label for the quantity of items in a set), but do we really want to do that? Isn't the value of a definition its ability to link our symbols to the real world, via experiences we can all share, so that we can all use the symbols for useful things?

Please do the operations in another video

wtfff excelentttt

So "successor" is a label attached to the steps of a recursive function.

To any programmers around, including Gabe, who think this topic is neat: consider reading / working through Software Foundations by Benjamin Pierce.

lol can't wait to hear about bertrand russell.

On the metalic ratio problem... What if that turn out to be equivalent to the Riemann Hypothesis? Because "huge list of things indicating a hypothes is true but no proof" sounds just like it

Is it possible to create a set of axioms that leads to a number system that works differently from the natural numbers?

@MalachiWadas

2 жыл бұрын

Yea, it’s pretty simple

Is absolute zero the kelvin version of the 5th axiom

Can you explain why this makes the natural numbers rather than another sequence, like even numbers or powers of 2?

@alexanderf8451

6 жыл бұрын

Well it makes a set that is exactly the size of each of those sets. Its not until you equip those sets with operations that they begin to look different. Actually they're still very similar if you do that. The natural numbers with addition, the even numbers with addition, and the powers of two with multiplication are all isomorphic! They are identical in structure!

At 5:38, he says that S(S(S(Z))) != Z, S(Z), S(S(Z)), if S(Z) = Mario, then the Mario mapping to Luigi and Luigi mapping to Mario should not exist in the set. Otherwise, the axiom 4 is violated.

Looks a lot like lambda-programming.

@empathogen75

6 жыл бұрын

MrDiarukia in fact church numerals are an equivalent formulation of natural numbers from lambda calculus

@ChurchOfThought

6 жыл бұрын

John Thompson Or combinator calculus

@JM-us3fr

6 жыл бұрын

Lambda calculus is a bit more complicated than Turing's machine

@CuulX

6 жыл бұрын

+Jason Martin, By what measure? What do you mean by "more complicated"? For a computer they are equivalent, they are both turing complete and no computer sees any "difficulty" in computing anything. Any computation described in lambda calculus can be compiled to the turing machine language and vice versa so neither has any inherent speed or memory requirement benefits.

How can natural number be building blocks for integers? how does it build negative integers?

Self evident means shifting emphasis? General methodology, overview: If the concept or idea of a particular identity is not inclusive of all probabilities, then the excluded possibilities have to be defined, because the context of QM-Time is continuous and complete in-form-ation, and formulae of the connection/rules of identification of all phenomena are unitary.

What is bad about cycle definitions?