﻿How are the numbers developed from sets?


Let’s assume an axiomatic system of ZFC, which means:
1. Axiom of epsilon relation
2. Axiom of the empty set
3. Axiom of pair sets
4. Axiom of union sets
5. Axiom of replacement
6. Axiom of powerset
7. Axiom of infinity
8. Axiom of choice
9. Axiom of foundation


The axiom of infinity guarantees that the natural numbers are a set.


So now you have the natural numbers (as a set). We want to define an operator + on the natural numbers.


We do this in the CS way, by first defining an add1 and sub1, and using those to define addition. In mathematics we define add1 as add1: x -> {x}. Essentially wrapping an extra bracket around. This is because of the way we define the natural numbers as being:


0 -> {}
1 -> {{}}
2 -> {{{}}}
3 -> …
…


So indeed this definition of add1 is quite cool


The definition of sub1 is essentially the same, but instead it’s sub1: x -> x element of x


This essentially removes a bracket around x, or subtracts 1.


Now we define n-addition using recursion. We say that subn is recursively sub


S^n :=  S   ◦ S ^ sub1(n) if   n  ∈ N


S^0 := id N


Using this n-addition, we define + as being:


 ( m,n )  →  m  +  n  :=  S^n (m)


We can define addition using a similar technique, using repeated addition.


Now we have the natural numbers with addition and multiplication.


Next what we do is use quotient sets to define the integers. The TLDR is that we will quotient up the space of tuples (n,m), where your fuzzy glasses can’t tell apart two tuples (m,n), (p,q) where m+p=n+q. It’s a nifty way of enforcing that m-n = p-q, without ever actually referring to some function minus.


The study of equivalence relationships are interesting, but the TLDR is that they must be:
1. Reflective
2. Symmetric
3. Transitive
All three of these conditions must be met for the relation to be an equivalence relation.
The funny joke is to give examples of almost equivalence relations, but that fail, and see where they fail. The example Schuller gives is purposefully ambiguous: x~y if x loves y.


Reflectivity holds if one loves themselves.
Symmetry holds if when one person loves another, the other loves the person back.
Transitivity holds if a loves b, and b loves c, then a loves c. This is only true in some French movies, the joke being that French movies have a lot of romance.


The requirements for equivalence relations is important because it makes sure that after doing an eventual fuzzy glasses, or classification, that everything is well defined, and you get a MECE.