I'm following Real Analysis: A Long-Form Textbook (Jay Cummings) and there is a part about defining the positive set $P \subseteq \mathbb{F}$.
The following definition is given:
An ordered field is a field $\mathbb{F}$, along with the following additional axiom:Axiom: There is a nonempty subset $P \subseteq \mathbb{F}$, called the positive elements such that
(a) if $a,b \in P$, then $a+b \in P$ and $a \cdot b \in P$; and
(b) if $a \in \mathbb{F}$ and $a \neq 0$, then either $a \in P$ or $-a \in P$.
However, I find the $-a \in P$ a little odd. Clearly this can't mean that negative numbers are part of $P$. Since suppose that $-1$ and $-2$ are in $P$, then (a) could not hold since $(-1) \cdot (-2) = 2 \notin P$. So what does that part of the definition serve? Is it so that we can define -(-2)? Or is this more obvious in a case where we are not talking about numbers (I assume there must also be an ordered field of for example functions or so?)