Eli5: How define arithmetic operations from scratch with set theory?
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Start with [Peano arithmetic](https://en.wikipedia.org/wiki/Peano_axioms). The *natural numbers* {0, 1, 2, …} can be represented as follows:
* 0 is a natural number.
* Every natural number N has a unique *successor* S(N) which is also a natural number.
You can read “S(N)” as “the successor of N”. So, 1 is the successor of 0, so it’s S(0).
This lets you represent the naturals like this:
* 0 is 0
* 1 is S(0)
* 2 is S(S(0))
* 3 is S(S(S(0)))
* … and so forth.
Peano arithmetic turns out to be enough to do quite a lot. You can derive addition and then multiplication from these basic axioms. Here’s addition:
* X + 0, or 0 + X, is just X.
* S(X) + Y is the same as X + S(Y).
2 + 3 is represented as S(S(0)) + S(S(S(0))). By applying the last rule repeatedly, you can move all the S’s over to one side, giving S(S(S(S(S(0))))). Which is 5.
Multiplication of natural numbers is just repeated addition, so we can construct that similarly:
* X · 0, or 0 · X, is just 0.
* S(X) · Y is the same as (X · Y) + Y.
Notice that these rules imply that X · 1 is the same as X. We don’t need to have a separate rule for multiplying by 1; that happens automatically because X · 1 is the same as X · S(0) which is the same as (X · 0) + X which is the same as 0 + X which is the same as X.
You can represent Peano arithmetic in any other system that gives you the concepts of “zero” and “distinct successor”. For instance, Peano arithmetic can be represented in set theory as follows:
* 0 is represented as the null set {}, also called the empty set.
* For every natural number N, its successor is represented as {N, {}} (the set whose elements are N and the null set).
Which gives these examples:
* 0 is {}
* 1 is {{}, {}}
* 2 is {{{}, {}}, {}}
* 3 is {{{{}, {}}, {}}, {}}
* … and so forth.
For any number, you can construct its successor by making a set containing the number and the null set.
Since we already know how to do addition and multiplication on Peano numbers, we can adapt that to work on these numbers too.
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