What Does It Mean to Be a Number? (The Peano Axioms) | Infinite Series

[@Frownlandia]

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]

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

[@enlightedjedi]

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]

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]

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

[@alexanderf8451]

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

[@enlightedjedi]

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

[@Cardgames4children]

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

[@tiana_roseee]

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.

[@NamelessP-wv6lf]

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!

[@AFastidiousCuber]

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]

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]

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]

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]

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

[@rufusneumann9703]

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,∞)

[@mohammedal-haddad2652]

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

[@fallenlegacyz]

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

[@talideon]

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

[@veggiet2009]

Gerard Tan touche

[@ocean_0602]

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]

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]

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

[@letstalkaboutmath2121]

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

[@alexwang982]

LetsTalkAboutMath Brains?

[@ThomasGodart]

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

[@MIsterremix99]

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]

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

[@gamechep]

I have been waiting for this video for years!

[@Tundra1428]

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.

[@shyamdas6231]

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

[@PPAChao]

Sponsored by Nintendo? ;)

[@pronounjow]

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]

How

[@Zurumpiyer9415]

😂😂😂

[@zhadoomzx]

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]

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]

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]

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]

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]

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)

[@alexnpe]

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

[@anon8109]

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.

[@pronounjow]

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

[@InShortSight]

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

[@sharonmurphy9973]

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.

[@michaelnovak9412]

That was awesome!!

[@robertkiss7525]

Prokopf approach: Axiom: we suppose (and we accept it) our universe to be real and definitly limitless. Consequences (main rules): 1. Every point could be a center point of the universe (and also it is). 2. The non-existence of something (which one is well definited) is unprovable. Only the conditional exclusion is provable. 3. The applied logic is always limited by the system of conditions (= logical space). All the results are true only inside the logical space. 4. Polarity theorem. Every thing which one exists must to have at least one basic property (essential feature) which makes difference between the "existence" and "non-existence" of the thing. To assure the "non-existence" every thing, it musts to exist the negated essential feature as well. These basic properties are called polarities. (Every other features are consequence properties.) 5. The value of something depends on the position of the evaluator at the eventline.

[@ricardopeano4147]

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

[@Xappreviews]

The 5th axiom doesn‘t make sense to me. If we have T = { 0,1,2,3, ... } U { Luigi, Mario }, then 0 is in T and for every x, s(x) is also in T. So then T fullfills the axiom. What am I missing? In my understanding, the axiom guarantees that every possible natural number is contained in T, but ist does not prohibit the possibility of two additional elements with a S-Loop

[@alexanderf8451]

Agreed. S(a) = b and S(b) = a definitely doesn't break the rule that S(x) always exists. I must be missing something.

[@dance1211rec]

What it is saying is that if we can find a subset of N then that subset must be N itself. If you have T in your case then {0,1,2,3,...} is a subset of T that satisfies 0 is in T and s(x) is in T if x is in T, but {0,1,2,3,...} is not equal to T. Therefore it is not N.

[@mcorony4727]

If I understand correctly, what the 5th axiom states is that if T is a subset of N, and it satisfies that 0 is in T and for every x in T, s(x) is also in T, then T is necessarily N. For the case of N = {0,1,2,...} U {Luigi, Mario}, we could take T = {0,1,2,...} and it would satisfy the condition, but T is not equal to N, therefore N doesn't satisfy the axiom

[@supercool19961996]

You should read the 5th axiom as a restriction on N, not as a restriction on T. If you take N = { 0,1,2,3, ... } U { Luigi, Mario }, then N has the subset T = { 0,1,2,3, ... }. Now T contains 0 and for every x in T, S(x) is in T. However T is not the same as N, as Mario and Luigi are not elements of T. The 5th axiom now says: Our choice of N is wrong!

[@Xappreviews]

Oh, yeah, the axiom must be true for ANY T that satisfies the conditions. Somehow my head interpreted „any“ as „at least one“. Thanks to all of you ;)

[@jasondoe2596]

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]

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

[@gabemoser1]

I think this is the best video on KZbin

[@pbsinfiniteseries]

While I appreciate the compliment, *this* is actually the best video on KZbin: kzbin.info/www/bejne/aqOaqHuDnJqMoMk

[@Pika250]

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.

[@jaimeduncan6167]

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

[@AlexBerg1]

Peano and Dedekind are great! Thanks for showing!

[@WRAWLINGSON]

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

[@davidlythgoe809]

best video on this topic... thank you!

[@azuliner6745]

I'm looking forward very excited for the next episode

[@rkpetry]

[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...

[@ChurchOfThought]

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]

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]

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 KZbin. 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

[@Blink-ut7uh]

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

[@anteconfig5391]

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.

[@broccoloodle]

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.

[@dom.ragusa]

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! :)

[@Avinash-wg7xl]

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

[@pirmelephant]

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?)

[@gianlucabasso]

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]

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]

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]

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]

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]

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

[@fejfo6559]

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

[@CuzicanAerospace]

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.

[@robchr]

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]

Oh, *yes* please!

[@_joncitone_]

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

[@cengelbey8743]

I don't understand axiom 4. Instead of these nested S function, let me use numbers. Consider S(0)=1, S(1)=2, S(2)=3. All these required by the axioms 1-4. Now S(3) can not be 2 by axiom 4 but what prevents S(3)=1. I get that i can not loop two numbers like Luigo and Mario but what happens if I create a loop with three numbers? Specifically I don't see why S(S(S(Z))) != S(Z) at 5:50.

[@pbsinfiniteseries]

Axiom 4 says two different inputs to S cannot give the same output, i.e. that S is 1-to-1. So what prevents S(3)=1, in your example, is that you already have S(0)=1, so nothing else is allowed to yield the output 1. So if you create a loop (and you could create a 3-loop, or a 4-loop, or whatever -- I just chose a 2-loop for ease of exposition), that loop has to be "detached" from all the stuff that you get through successive S-ing of 0 and its successors.

[@Nixitur]

Good question, it took me a bit of thinking, too. But here's why: The fourth axiom states that if S(x) = S(y), then x=y. So, if S(S(S(Z))) = S(Z), then, in this example, x=S(S(Z)) and y=Z. By axiom 4, this would mean that S(S(Z)) = Z. However, axiom 3 states that Z can never be the output of S. So, S(S(Z)) can not be Z. So, S(S(S(Z))) can't be S(Z) either.

[@翰-u6z]

I've got the same question, and you guys explained it so well. Thanks!

[@cengelbey8743]

PBS Infinite Series Thanks. Now that you explained it seems so obvious.

[@kennarajora6532]

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.

[@shriramashirgade2571]

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.

[@ronitbhatia4783]

Can't figure out why axiom 5 is needed ? Isn't axiom 4 sufficient to prevent the Mario - Luigi loop from existing ? S(S(Mario)) = S(Luigi) = Mario, which is explicitly forbidden by axiom 4?

[@MikeRosoftJH]

Axiom 4 only prohibits two different numbers having the same successor. Because S(Mario)=Luigi, that S(S(Mario))=S(Luigi) doesn't contradict anything. (To the contrary, that follows from the definition of equality in mathematics.)

[@NiallsSongs]

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

[@MrRyanroberson1]

7:00 you replied to my comment with EXACTLY what I said. You used faulty logic, since already with axiom 3 s(Luigi) CANNOT be Mario, therefore that is a BAD example of why you might need an additional axiom to minimize the sey

[@pbsinfiniteseries]

I think you are misunderstanding either axiom 3 or axiom 4 or both. None of the axioms say "Applying S twice in succession to any given element X cannot give you back the same X you started with." None of them say that. So let's see, then, where we get that S(S(Z)) cannot equal Z.... When we have only axioms 1 and 2, nothing stops this. We know Z is in ℕ at this stage b/c axiom 1 says so. S(Z) has to be in ℕ by axiom 2, but we don't know at this stage that it's different from Z. Suppose it isn't different from Z, i.e. that S(Z) = Z. Then S(S(Z)) also = Z, as does S(S(S(Z))), and so forth, so we have only a single element Z in the set. Suppose that S(Z) is different from Z. Then still nothing stops S(S(Z)) from being equal to Z, giving a two-element S-loop with only 2 elements in our set. We want to prevent both those scenarios. So we introduce axiom 3, that Z is never the output of S. Now both the aforementioned possibilities are blocked, b/c both require Z to be the output of S at some stage. Note that axiom 3 applies *only to the element Z* mentioned in axiom 1. It will not apply to Mario later, since Mario isn't the same thing as Z. More specifically, there is no rule anywhere that will say "Applying S twice to an arbitrary element X is not allowed to give you back X." That's never going to be a rule (until axiom 5 ends up implying that indirectly by outlawing S-loops), and it was never the reason that S(S(Z)) was prohibited from equaling Z. Anyway... so how many elements are we *guaranteed* to have in ℕ now? We could have an infinite chain, yes, but we are still *guaranteed* only two elements -- Z and S(Z). Why? Let's look at the behavior of S on each element we have so far. Z is there by axiom 1. S(Z) is there by axiom 2 and isn't the same as Z, so that's a legit 2nd element. What if we feed S(Z) into S? By axiom 2, the result S(S(Z)) must be in ℕ, and it can't be the same as Z (by axiom 3). But it *could* be the same as S(Z)! In other words, we could have a one-element S-loop that S(Z) forms with itself, if the rule for S "behind the scenes" were really "The output is always S(Z)". So we need to stop _that_ . Enter axiom 4 -- you can't get the same output from two different inputs. Now S(S(Z)) has to be different from S(Z) (it already had to be different from Z by axiom 3). Likewise, S(S(S(Z))) is different from Z (by axiom 3) and from all other known outputs of S so far (by axiom 4). Ditto as you keep S-ing over and over -- you have to keep getting outputs that aren't already in your list, so your set ℕ now has to have infinite size. But still nothing stops us from having two other elements M & L that *aren't part of that infinite chain at all* . They can form a two-element "island" unto themselves if S(M) = L and S(L) = M. This doesn't violate any of the axioms 1 through 4. Watch... Is Z still in my set? Sure, so axiom 1 checks out. Is the output of S still always in the set? Sure -- the output of S(M) is L (which is in the set), and the output of S(L) is M (which is also in the set). Thus, axiom 2 checks out. Axiom 3 checks out b/c Z still isn't ever the output of S. And axiom 4 checks out b/c outputs of S are still *unique* . *This* is why you need axiom 5: to guarantee that every element of ℕ can be arrived at by moving either down or up the successorship chain no matter where you decide to initially situate yourself along that chain. Without axiom 5, you could start at, say, S(S(Z)) and never arrive at either M or L no matter how many times you applied S or "undid" S, b/c M & L wouldn't be part of your infinite S-chain. They'd live detached, in an S-loop unto themselves. Axiom 5 is an extra rule that outlaws those detached S-loops. Get it?

[@abhik294]

Was searching for this same question and this awesome reply.......Thanks to both Ryan and Sir Gabe 😊😊😊😊😊😊😊😊

[@Wheelrezz]

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

[@rhyno3780]

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

[@ferdinandkraft857]

Great job, Gabe!

[@Camboo10]

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?

[@JM-us3fr]

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]

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]

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]

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

[@matthewleitch1]

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?

[@georgeentertainment7185]

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.

[@steliostoulis1875]

You will be missed ❤️

[@Flatunello]

So where does Peano fit in after Godel had his way with the foundations of arithmetic?

[@Icenri]

It seems to be an open question. Some people were talking about first and second order Peano arithmetic. I'm reading that Hilbert's second problem was the one Gödel solved, proving first-order Peano arithmetic inconsistent but after that efforts were made (none of which I'm able to understand) as we can read here: en.wikipedia.org/wiki/Peano_axioms#Consistency Still there's modern stands saying that Hilbert's second problem is still unsolved as is stated here: en.wikipedia.org/wiki/Hilbert%27s_second_problem Finally, what it seems mathematicians use today is ZFC to make Peano arithmetic consistent as I read here: en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#Consistency

[@rosaliedefourne725]

Yes, there are proofs that Peano's arithmetic is consistent. I think Gentzen was the first to provide a proof, using sequent calculus.

[@markus9871]

There are Goodstein sequences which can be formulated using Peano Axiom but you can't proof their convergence in it. On the other hand their is the Presburger Arithmetic which is weaker then Peano but complete and decidable. Meaning not every axiomatic system conatin undecidable statements.

[@Flatunello]

It was a question to stir up a little conversation. Godel showed that all mathematical (including systems that could be put in mathematical form) are either inconsistent or incomplete resulting in results that often were not certain. He wrote about Russell and Whitehead's Principia Mathematica, a formalization of the basis of arithmetic and showed the theory incomplete. It was a stunner then and still is. Mathematicians fear some conjectures may never see proofs that are nevertheless true. Here is a link: kzbin.info/www/bejne/hWXRlXx6mKmGfcU

[@Deguiko]

If you consider the 5th axiom (of induction), the way it is stated in the video, then, Godel's theorem doesn't apply. Godels theorem says that many different objects (number systems) will satisfy your axioms for arithmetic, as long as you write your axioms in the language of First-Order logic. If two number systems are different, some statements will be true in one and not in the other (the axiomatic theory is incomplete). What makes the induction axiom so different is that it states something about ALL SETS of natural numbers, which can only be writen in 2nd Order Logic. To illustrate: -All ELEMENTS are such that... (First Order statement) -All PROPERTIES of elements are such that... (Second Order statement)

[@Mustachioed_Mollusk]

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.

[@MrDiarukia]

Looks a lot like lambda-programming.

[@empathogen75]

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

[@ChurchOfThought]

John Thompson Or combinator calculus

[@JM-us3fr]

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

[@CuulX]

+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.

[@192ali1]

Excellent lecture. Thank you

[@RasperHelpdesk]

A bit confused on how axiom 5 excludes the Mario-Luigi chain from T, from the rules at 7:20 T contains s(Mario) if T contains Mario => T contains Luigi if T contains Mario likewise T contains s(Luigi) if T contains Luigi => T contains Mario if T contains Luigi So T contains both if T contains either, so doesn't that mean T can contain both? Or is the trick that neither Mario or Luigi ever get added first, and therefor neither get added at all?

[@Xappreviews]

Exactly my thought! To my understanding, the 5th axiom only says that all successors of Zelda are included in the natural numbers, but does not exclude the S-Loop for two more elements. So the peano axioms allow us to assume that luigi and mario are in N

[@Xappreviews]

I read the wikipedia article in english and german, and both do not say that the 5th axiom guarantees minimality

[@pbsinfiniteseries]

Hey all, check out the response by +Dance1211 to another comment (linked here -- kzbin.info/www/bejne/aZiloINun79kf9U&lc=UgzPm0pc2gVeXnasKHd4AaABAg.8dBKRnsJndK8dBN9luNPF- ) . It summarizes the explanation nicely.

[@lordcublington]

Peano’s first axiom is “0 is a natural number." How, then, can we say that the Peano axioms define the natural numbers? The Peano axioms do not define the natural numbers. They assert a successor relation, and the natural numbers relate to each other in this way. Axioms do not define.

[@bawol-official]

IMO the concept of zero has always seemed more like a state of existence more than it signifying any quantity. Nothing isn’t a form of measurement until you introduce elements into a set. 0 is the “on” light for any mathematical function.

[@KatTallest]

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]

Waddup! Kelsey went on to finish her PhD. Tai-Danae Bradley have taken over the channel as co-hosts. This was all announced here: kzbin.info/www/bejne/hHnYlKOwl6lpZpI

[@abhik294]

Enlightening Video Sir🙏🙏💥💥💥

[@davidwilkie9551]

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.

[@ReidarWasenius]

THANKS for another great video!!

[@tothespace2122]

At 4:55 axiom 3 states that Z can never be an output of S but on the right side of the screen we see the equation S(S(Z))=S(Z) which is allegedly allowed by the axioms but can't that equation be true only if S(Z)=Z (which contradicts axiom 3), thus S(S(Z))=S(Z)? I am stuck here. I can't see how this is allowed even though axiom 3 states Z can never be output of S. I would be thankful if someone could explain this a little bit.

[@MikeRosoftJH]

Okay, let's take a two-element set {0,1}, where S(0)=1, and S(1)=1. We have so far only posited that S(n) is not equal to 0 for any n, which is indeed the case for our set. That's why we have added another axiom: if m is not equal to n, then S(m) is not equal to S(n) (which doesn't hold for our model: S(0)=S(1)).

[@Ifslayanct]

We miss you over at Space Time.

[@RichardLightburn]

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

[@fawzibriedj4441]

Nice episode ^^ thanks and keep up :)

[@ahmidii]

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?

[@karabomothupi9759]

Man. This is some crazy stuff

[@HoD999x]

gabe: "can you explain natural numbers without using circular definition?" me: "yes! just take an infinite set and pick one element at random. that we call zero. now pretend that zero has a next element (other than itself). from then on, every element has a "next" element that is among the set of not already used elements. tadah!" gabe: "you can't use next" me: "but...but...." gabe: "ready? now let's define next....." me: "but this is exactly what i just said...." :|

[@pbsinfiniteseries]

You're welcome ;)

[@isaacc7]

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?

[@bojankostic2403]

Welcome back

[@tsunghan_yu]

Wow so well explained

[@kylebowles9820]

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

[@JakubH]

I still don't understand how the axiom 5 gets rid of the mario-luigi (or other) cycles, and of chains starting from a totally different element successor-ing forever. If I put T=thatweirdN, then all the requirements for being element of T hold, so we have a set weirdN with cycles and other chains, which was then declared to be the N, but it doesn't look like that...

[@MikeRosoftJH]

Let T[x] be the property: x≠S(x) (x is not a successor of itself). 1) T[0] is true (because 0 is not a successor of any number). 2) if T[x] is true, so is T[S(x)] (because two different numbers can't have the same successor). 3) By axiom of induction, T[x] is true for all x. We can instead take T to be the statement "x is not a part of a cycle of length n"; that is, "x≠S(S(x))", or "x≠S(S(S(x)))", or (and so on). The proof will be similar: T is not true for 0 or 1 (and so on) because that would contradict that 0 is not a successor, and the induction step is true because otherwise that would contradict that two different numbers can't have the same successor. We can also prove by induction the more generic statement: if y≠0, then x+y≠x.

[@jenniemaes1967]

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

[@bielczas]

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.

[@onemikeb]

I am confused... The minimal subset of {Z, S(Z), S(S(Z)), ...} U {M, L} is {M, L}. {M,L} is smaller than infinity and it satisfies all of the axioms.

[@freediugh416]

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

[@saarrrcamscms226]

Love love this !

[@3ckitani]

I think that numbers are words that we use to define how many things are there, and then after we messed up with fractions, exponentials, and other things, it suddenly become an abstract concept.

[@zetadroid]

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

[@romajimamulo]

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

[@chandankar5032]

Please ! Can any one tell me the need of axiom 5, i did not get that mario-luighi thing. 6:36 how that set bigger than N ?

[@MikeRosoftJH]

The S (successor) function is essentially a "+1" operation (though this image is really backwards - the operation + and the number 1 are defined using the successor function, not the other way around). And we don't want to have numbers a and b such that a+1=b and b+1=a.

[@sugarfrosted2005]

I feel like the explanation of the axiom of induction you used misrepresents what it actually implies. The Peano axioms allow for non standard models with larger models than the naturals. Like having an element that's bigger, it just has to have infinitely many predecessors.

[@pbsinfiniteseries]

Fair, but what I meant to get across is the axiom of induction is there to prohibit "successorship loops" that are detached from the straight "successorship chain" that characterizes inductive sets.

[@guitarraccoon1541]

So if we can't use any numbers at all, can't we define 0 as the cardinality of the null set and go from there? Or are we not allowed to use cardinality at all.

[@pbsinfiniteseries]

No cardinality. That's a concept that you need to define in terms of numbers. Interestingly, you can end up defining equinumerosity without numbers, but to distinguish among the different equivalence classes generated by the relation of equinumerosity, you need something that is tantamount to numbers. I make a minor comment about this in the follow-up to this episode (which airs tomorrow).

[@nikolaichow4663]

Really helps! thanks!

[@EugeneChangEC]

I can't quite draw a clear line between what was shown and how it proves that natural numbers exist. It seems to prove that a "sequence" of things exists, but why does that sequence have to be the natural numbers? Is it because you can number the iterations? Is there something else?

[@pbsinfiniteseries]

It doesn't prove that the naturals exist. Rather, it distills down to basics what it is about the familiar set of natural numbers that gives it its unique character. At least in my viewpoint, the Peano axioms are a *characterization* of the natural numbers, not a *proof of existence* of the natural numbers. Not that we necessarily need to prove the existence of the naturals -- I mean, we can all count, and we're clearly doing _something_ when we do so.

[@AngryArmadillo]

Will the next video be on ZFC?

[@iwersonsch5131]

A sizeable part of the world sets Zelda to be 1 rather than 0, the natural numbers still work. The only difference I see is that addition and multiplication will have to be defined a tiny bit different: Add (x, Zelda) := S(x) Mul (x, Zelda) := x

[@JoakimfromAnka]

This guy should come back to Spacetime.

[@shaylempert9994]

Wonderful! Thank you!

[@AJBlue98]

Okay, let's see if I can get this right from memory... There exists a set ℕ and a function S. All members of ℕ are inputs to S. ℤ⃒ is a member of ℕ but is never an output of S. All other members of ℕ are outputs of S. No input to S shares output with any other member of ℕ, and no output from S shares input with any other member of ℕ. Do I have that about right?

[@pbsinfiniteseries]

Yeah, except for two things: (1) You don't need this statement: "All other members of ℕ are outputs of S." That will turn out to be guaranteed by everything else. (2) You need to enforce separately that ℕ is the *minimal* set that will work like this (i.e. axiom 5).