I am considering the following problem: Is the subset of positive real numbers unique, according to the Positivity Axiom (cf: Page 8, Fitzpatrick, Real Analysis, 4th edition):
... There is a set of real numbers, denoted by $\mathcal{P}$, called the set of positive numbers. It has the following two properties:
P1 If $a$ and $b$ are positive, then $ab$ and $a+b$ are also positive.
P2 For a real number a, exactly one of the following three alternatives is true: $a\text{ is positive, $-a$ is positive, $a=0$.}$
In order to prove the uniqueness of the subset of positive numbers, I have tried like this: Assume there are two subsets of positive numbers, denoted by $\mathcal{P}_1, $ and $\mathcal{P}_2$, then I have proved that $1\in\mathcal{P}_1\cap \mathcal{P}_2$, then I do not know what to do. Can anyone help me?