How can a statement like “5 is greater than 4” be quantified? Or is it just obvious? If we know anything about mathematics we know that there is no way we can assume something as obvious, there must be an axiomatical contruct that puts a rigorous meaning on “5 is greater than 4“.

The first attempt would be to say that 5=4+1 so 5 is “1 more” than 4 so must be bigger. This translates to 5-4=1: “5 is greater than 4 because 5-4 is positive“. Careful! 4=5+(-1) so 4-5=-1: “4-5 is negative“. But what does positive and negative mean? Easy? Positive is greater than zero… At this point a stronger construct is needed:

Definition: Call a set $P\subset \mathbb{R}$ positive if for all $a,b\in P$

1. $a+b\in P$
2. $ab\in P$
3. Given $x\in \mathbb{R}$ either $x\in P$, $-x\in P$ or $x=0$

If we think carefully, this definition concurs exactly with that of the naive notion of positive. So we can say that “5 is greater than 4 because 5-4 is positive.”

Definition: Given $a,b\in\mathbb{R}$,  $a$ is said to be greater than $b$, $a>b$, if $a-b$ is positive.