Is the condition $x\in\mathbb{R}$ necessary to the set statement $\{x \in\mathbb{R} \vert x> 0\}$? Forgive my ignorance.
Is the condition $x\in\mathbb{R}$ necessary to the set statement $\{x \in\mathbb{R} \vert x> 0\}$?
In other words, if $x$ is greater than zero, then is it not, by definition, a real number?
Thank you very much!

It's absolutely necessary. You could have for instance
$$\{x \in \mathbb{Q}: x>0\} $$
which only includes positive rationals, not the irrationals.
It's common to abbreviate these to $\mathbb{R}_{>0}$ and $\mathbb{Q}_{>0}$ respectively.
