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As previously explored, we can define the natural numbers in terms of sets. We can even use them to count. But how are operations like addition and multiplication defined? Along with a crash-course in proof by induction, we'll define these operations in this video and prove that they possess all the lovely properties with which we are familiar.

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00:00 - Intro
04:02 - Axiom Recap
05:28 - First Attempt at Addition
09:05 - Addition Definition
15:30 - Inductive Principle
21:45 - Associativity of Addition
25:25 - Commutativity of Addition
30:30 - Multiplication Definition
37:50 - Distributivity
43:09 - Subtraction?
44:25 - Closing Remarks

*It is sufficient to define addition without the n+1:=S(n) part. However, I made the pedagogical choice to include this -- it's less efficient but more intuitive for the uninitiated, in my opinion.

**The base cases of our inductions should deal with the n=0 case, as then the n=1 case is true by virtue of "true for k ⇒ true for (k+1)" where k=0. However, the n=0 case is always completely trivial so I decided to make my base cases at n=1 so that viewers less experienced with proof by induction get to see more "work" being done and get a feel for how to complete proofs by appealing to axioms. Let's just go ahead and assume we've proved the n=0 case separately in all proofs!

***I misspeak here: We are proving that if a given number k commutes with *a* then the next number commutes with *a*. Apologies!

****Psst, it's me, Yellow T-Shirt Alex, put the solution in the website slug. You'll also need those numbers I told you to keep safe!

All music by Danijel Zambo.

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