Clarification on The Trichotomy Law I'm getting this from Spivak's, "Calculus" 4th ed. Pg 9. If $P$ is the collection of all positive numbers, how is $-a$ in the collection $P$? Does it mean for some positive number $b$ there exist $b-a$ that is positive and thus $-a$ is in $P$? But that doesn't sit right with me.

If $P\subset \mathbb R$ is the set of positive numbers then ($P$ is closed under addition ansd multiplication and) $\mathbb R$ is the disjoint union of $P$ , $\\{0\\}$, and $-P$, that is for each $a$, either $a=0$ or $a\in P$ or $a\in -P$ and the latter just means that $a$ is negative and th epositive number $-a$ is in $P$.