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See how we develop even more concepts from this mathematical foundation.

Ever wondered how numbers are actually defined? In this video, you'll learn the most common way it's done by mathematicians.

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00:00 - Intro
02:19 - Motivation
04:12 - Physical Units: An Analogy
08:26 - First Attempt
10:39 - The Perils of Intuition
12:40 - Definition in Principle
15:38 - Sets and Russell's Paradox
23:44 - Rules of Sets
31:46 - Constructing the Numbers
38:25 - Cake
38:29 - Closing Remarks

*Galileo's Paradox: In his final book, which bears the catchy title, "...," Galileo discusses the size of the set of natural numbers versus that of the set of square numbers (not even numbers, as in my video, but the point is the same). It's written as a dialogue, not as a letter as I erroneously claimed.

**Formulations of ZF: There are several equivalent ways of formulating ZF set theory. Some have that the empty set exists as an axiom, but not all. Others prove the existence of the empty set by using the axiom of infinity then the axiom of subsets, and others argue similarly to how I argue in that in first-order logic, something exists. The point is, I shouldn't have claimed that the axiom of the empty set is always part of ZF.

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