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

Video Description

Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: https://to.pbs.org/donateinfi 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? Tweet at us! @pbsinfinite Facebook: facebook.com/pbsinfinite series Email us! pbsinfiniteseries [at] gmail [dot] com 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. 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 Made by Kornhaber Brown (www.kornhaberbrown.com) 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!