Consider the following definition:
To say that P(x) is true for every value of x in the universe of discourse U, we will write $\forall x P(x)$
My question is, if I want to specifically state what the universe of discourse is, how do I express it mathematically?
For example:
$\forall x P(x)$, where the universe of discourse is the set of all positive integers.
How do I express "where the universe of discourse is the set of all positive integers" using mathematical notation?
If $\mathbb N$ denotes the set of positive integers then you can use the expression:$$\forall x\in\mathbb N\;P(x)$$or - if you prefer an unbound quantor - the expression:$$\forall x\;[x\in\mathbb N\implies P(x)]$$
Caution: many mathematicians (me too) use $\mathbb N$ to denote the set of nonnegative integers and in order to avoid any confusion you could also use $\mathbb N^+$ or $\mathbb Z_{>0}$.