Understanding What It Means to Be a Number: The Peano Axioms Explained

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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
https://youtu.be/KZT5hrYOERs

Crisis in the Foundation of Mathematics
https://www.youtube.com/watch?v=KTUVdXI2vng

How to Divide by "Zero"
https://www.youtube.com/watch?v=uxpowBoPieQ

Beyond the Golden Ratio
https://www.youtube.com/watch?v=MIxvZ6jwTuA

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.

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