What is a number ?

Construction of Natural Numbers

In this, I rigorously define the concept of a natural number, using Peano's axioms. I also explain why those axioms are the basis for the principle of mathematical induction. Enjoy!

[ragokyo]

"Because the axioms are so *instrumental* to our knowledge of math, we'll call it the Peano (piano) axioms." - Dr. Peyam 2020

[sexyyoda]

Dr. Peyam : What is a Number?
Me : *at 1am*
Yeah, I should definitely check it out

[liminf]

I am a Japanese girl and also, an university student.
I’m bad at English, but your lesson is so interesting, so I draw in your lesson.
I specialize in Lebesgue integral, I look forward to hearing from you about the topic someday.

[maxreenoch]

8:00
Dr. Peyam's Board: 3=1+1
Me: ...


A short while later...
Dr. Peyam's Board: *blank*
Me: Phew!

[blackpenredpen]

I once in a video said 2 is greater than 3 ...


I guess I was right.

[ayoubnouni]

Not even a month ago, i was looking for the whole construction of N and Z. This is quite interesting 🧐 🙏🏻

[martinepstein]

I like to compress the first 4 axioms into "There is a natural number 1 such that there is an injection s: N -> N - {1}". This is logically equivalent to saying N is infinite but you might as well introduce the symbols "1" and "s" while you're at it.

[bugrahankarahan]

Peano axioms has been changed .it allows that 0 is in Natural numbers

[janvandergaag]

Nice video. Though I still have one concern. When not including 0 as the first element of N, why start at 1? The set {3, 4, 5, 6, ... } also complies to all 5 axioms. Where 'successor of' is not +1 of course, because addition has not been defined yet. Just that 3 HAS a successor, call it 4; etc.) For all we know, the element labeled '1' in the video really has the value of 3. The numbers in the video don't have any other properties (yet).
Or consider the set {3, 6, 9, 12, ... }, multiples of 3, and 'successor of' means something like 'add the smallest number in the set'. This also complies to the axioms, again with different labels assigned to individual numbers, and having actual values different from intended.

If we are to include more properties, like addition, we could define 0 as the element such that n + 0 = n for all n in N. (I prefer 0 to be included in N.)

[Hello_am_Mr_Jello]

thanks for this playlist, Am gonna watch all of it...

[gast]

Good explanation besides that I have also learned now (through Wikipedia) that some define the natural numbers without the zero.

[iabervon]

"You might have no piano. You could always get a piano. If you got a piano, you wouldn't not have a piano. If you and someone else don't have the same number of pianos now, you wouldn't have the same number if you each got one. If people without pianos are in your group, and if people in your group stay in your group when they get pianos, then everybody's in your group."

[michalbotor]

actually i will be a little picky here and insist that we do not denote the successor of n by "n + 1", but by "s(n)" as it is usually done. after all what is addition of natural numbers anyways? why is the successor of n supposed to be 1 bigger than n? how do we know that the successor of n is the next n (no, the name "successor" is actually premature: it is named after what it is before we actually prove that it is)? what does bigger even mean? oh, boy!! we would actually have to define all of these first starting from nothing but these 4 axioms and 1 axiom schema (induction represents frankly infinitely many axioms in disguise). you see, once you really give it a thought and throw away all of your previous presumptions that you took from school it's actually bloody difficult to infer all of these "obvious" properties of natural numbers.
btw. peano axioms don't give you one natural numbers they give you infinitely many natural-like numbers. sets like {1, 2, 3, ..}, {0, 1, 2, 3, ..}, {.., -3, -2, -1} and {.., -3, -2, -1, 0} and even ones like {4, 5, 6, ..} and {.., -6, -5, -4} or the one by von neumann that dr peyam gave all pass peano axioms provided that one is abstract enough. i don't know how they are called in english, but in my country we call these "models of peano axioms".
and these axioms would go something like this:
we assume the following 3 root notions: that of set "Σ" (that's our set of natural-like numbers), element "a" and function "φ".
we assume the following 4 axioms:
1) element "a" is a member of set "Σ";
2) if element "a" is a member of set "Σ", then so is the element "φ(a)";
3) there exist no such element "s" in "Σ" such that "a = φ(s)";
4) for all elements "s" and "t" in "Σ", if "φ(s) = φ(t)" then "s = t"'
and finally we assume the following axiom schema:
5) if "S" is a subset of "Σ" such that
a) element "a" is the member of "S";
b) if element "s" is a member of set "S", then so is the element "φ(s)";
then "S = Σ".
i don't think that this approach is any harder than the usual one, but it's more powerful and what's most important it doesn't let us fool ourselves that "it's so easy, and there is nothing really to prove" so easily.
lastly, i highly highly recommend the coverage of this topic by the pbs infinite series channel (rip infinite series ;c, i miss you)

[juanfransanchez]

Still connected. We'll see how far I can get. Thanks for sharing this course with us.

[aneeshsrinivas]

why define the peano axioms for the natural numbers as opposed to say the integers.
is it possible to do a peano like construction for the integers directly?

[plaustrarius]

Very interested about defining 'the next number' in the reals, as Dr. Peyam was saying

We can use limit expressions to come up with numbers infinitely close to 1, on either side.

Is this was dedekind cuts and cauchy sequences are getting at? I can split the number line, get infinitely close to the limit points, and the cut or space is the number. Is that close?

If there is a systematic was of building these cuts cant we define the smallest number bigger than one somehow? I felt this was the spirit of the surreal numbers but not sure.

Thank You Dr. Peyam!

[cherylprell]

I've always had Good Luck with 9 and 20 and

[martinepstein]

I can't believe this. In his latest video 3blue1brown lists a few operations defined by iterating simpler operations: tetration is iterated exponentiation is iterated multiplication is iterated addition. I commented that addition is iterated successorship and now I see you're doing a series on Peano arithmetic! I even told a commenter who thought the name "successorship" was too fancy that they should come back after studying Peano arithmetic and say that (hopefully they heard me from all the way up on my high horse).


The situation at the end is even worse than you describe. Axioms 1-4 don't establish any kind of order relation at all (and even with axiom 5 this takes some work) so "least element" is just nonsense! As I mention in another comment axioms 1-4 are satisfied by literally every infinite set.

[seplhds]

can't we use well ordering principle to prove induction axiom?

[jakubfrei]

Thank you for this video ! May i ask you to do more vids about number theory ?