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

If you needed to tell someone what numbers are and how they work, without using the notion of number in your answer, could you do it?

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Previous Episodes:
Telling Time on a Torus

Crisis in the Foundation of Mathematics

How to Divide by "Zero"

Beyond the Golden Ratio

Are the natural numbers fundamental, or can they be constructed from more basic ingredients? It turns out that you can capture the essence of numberhood in a small set of axioms, analogous to Euclid’s axioms in geometry. They will allow us to build a set N that will behave just like the natural numbers without ever explicitly mentioning numbers or counting or arithmetic as we do so. These axioms were first published in 1889, more or less in their modern form, by Giuseppe Peano, building on and integrating earlier work by Peirce and Dedekind.

Written and Hosted by Gabe Perez-Giz
Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Mike Petrow and Linda Huang

Special thanks to Roman Pinchuk for supporting us on our Converse level on Patreon.

Along with thanks to Matthew O'Connor, Yana Chernobilsky, and John Hoffman who are supporting us on Patreon at the Identity level!

And thanks to Mauricio Pacheco who is supporting us at the Lemma level!

[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.

[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 :)!

[Cardgameschildren]

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

[letstalkaboutmath]

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

[NamelessP-wvlf]

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!

[fallenlegacyz]

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

[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.

[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.

[mohammedal-haddad]

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

[MIsterremix]

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

[ThomasGodart]

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

[shyamdas]

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

[jasondoe]

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?

[gamechep]

I have been waiting for this video for years!

[pronounjow]

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

[alexnpe]

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

[Tundra]

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.

[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?

[WRAWLINGSON]

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

[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?