I'm a bit confused by the rules of writing in mathematical notation.
In general, if I wrote
$\forall x \in (0,3)$
Is it implied that the possible values of $x$ are only real numbers, or could the above line allow for x to equal an imaginary number? In other words, is the above line the same as saying the following?
$\forall x \in \Bbb R$ $ s.t. 0<x<3$
Two parts to this answer.
First, in an expression like
$\forall x \in (0,3)$
there is always an assumption about the universe - the values of $x$ that may or may not satisfy the condition. If there's any ambiguity, the author should tell you explicitly what she's considering. In this case it's pretty clearly the set of real numbers (or perhaps the rationals, depending on the context). You have nicely rewritten the statement to make the universe explicitly $\mathbb{R}$.
Second, whatever the universe is in this case, it can't be the complex numbers, because inequalities like $x < 3$ make no sense there.
