Given that a positive cone is $$\forall x,y \in P, x+y \in P$$ $$\forall x,y \in P, xy \in P$$ $$\forall x \in F, \space \text{either} \space x \in P \space \text{or} -x \in P \space \text{or} \space x=0$$ where we define $$b-a \in P \iff a<b \space \text{for some}\space a,b \space \text{in} \space F$$
I understand that for the order field of real numbers, the positive cone is $(0, +\infty)$ and clearly $\mathbb{R} = (0, +\infty)$ is false but I'm not sure that I understand how to express this formally.