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!
 A: 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.
A: It's definitely necessary. If it weren't, that would imply that
$$\{x : x>0\}$$
would always denote $\{x\in \mathbb{R}: x> 0\}$ but obviously this won't always be the case since it makes sense to write
$$\{x\in S: x>0\}$$
for any ordered set $S$ that includes $0$.
A: One more point. It might not even make sense to say $x>0$ unless $0$ and $>$ are defined. 
For example the set $\{ x \in \Bbb C : x >0\}$ doesn't make sense as $>$ is not defined (consistent with addition and multiplication) on $\Bbb C$. 
Or even worse, consider the set of all states in the USA. Does asking if a state is "greater than zero" make sense? 
A: One might say: For me, $\aleph_0>0$ and $\aleph_0\notin \Bbb R$, so the restriction $x\in \Bbb R$ is necessary

But we may also view this from a different perspective, namely that the form of the expression tells us directly that we are dealing with a set (and not a proper class: We use the notation (also know as class-builder notation)
$$\tag1\{\,x\mid \Phi(x)\,\} $$
to denote the class of all objects $x$ that make the predicate $\Phi$ true. In general, such a class  need not be a set.
In the wide-spread Zermelo-Frenkel set theory, there are two important axioms (or rather axiom schemas) that postulate that certain sets exist in terms of (other sets and) predicates: The Axiom Schema of Replacement and the Axiom Schema of Separation (or Comprehension). The first says that given what is called a class function $F$ and a set $A$, there exists also a set that has as elements precisely all things that equal $F(x)$ for some $x\in A$. We commonly use the notation
$$\tag2\{\,F(x)\mid x\in A\,\} $$
(or perhaps $F[A]$) for this. Likewise, the Axiom Schema ofSeparation tells us that for every set $A$ and predicate $\Phi$, there exists a set that has as elements precisely those elements of $A$ that fulfill predicate $\Phi$. We commonly use the notation
$$\tag3 \{\,x\in A\mid \Phi(x)\,\}$$
for this set. 
So whenever we encounter something that looks like $(2)$ or $(3)$, we can immediately be confiendt that it is a set. With $(1)$, we'd have to stop and check -  and why would you as an author want your readers to do that unnecessarily?
