What is the scope of a variable in set-building notation? Let's take a set builder notation, if I define a set {$x$|$x$∈N} Is the scope of the variable supposed to be limited to the set-builder notation?
Would using $x$ again imply that the second use of $x$ is new variable, and different to the first, because the scope of $x$ in set builder notation is limited to it?
Is it more formally correct to define new variables for use in set-builder notation? Would this imply that re-using symbols is a sort of abuse of notation?
 A: The variables inside the set definition are dummy ones.
For instance if you define the set $S=\{a\mid \exists x\in\mathbb N, a=x^2\}$
Then $x\in S$ means $x$ is a square integer, i.e. it is actually the '$a$' from the inner statement, and not the '$x$' of the same statement.
Of course it is always preferable to keep consistent notations for understanding, same as you would probably rather not work with the complex number $y+ix$ (unless you are masochistic). It's not forbidden, but so error prone...
A: In logic, set builder (or, set abstraction) operator is syntactically a variable-binding term operator.
Variable-binding term operators constitute a class by themselves: They form terms from terms like functions, but differ from them in that functions do not bind variables. Like quantifiers, variable-binding term operators bind variables, but differ from them in that quantifiers operate on formulas, not on terms. Russell's definite description operator $\iota$ (originally, inverted iota) and Hilbert's $\epsilon$ operator are examples from logic. See also the Wikipedia article for mathematical examples.
The usual notation for set builder/abstraction operator is $\{\;\mid\;\}$ or $\{\;:\;\}$. In order to display its logical form, it may be helpful to represent it as $\sigma x\phi(x)$ in which $\phi$ is a formula defining the comprehension of the set. Thus, for example,
$$\{x\mid x\in\mathbb{N}\}$$
can be represented as $$\sigma x(x\in\mathbb{N})$$
The occurrences of the variable $x$ have distinct syntactic roles. In the first occurrence, $x$ is a constituent of the operator prefix $\sigma x$, and in the second occurrence, it is a constituent of the matrix $(x\in\mathbb{N})$, which is an open formula.
Hence, $\sigma y(y\in\mathbb{N})$, $\sigma z(z\in\mathbb{N})$ (and so on with other variable symbols in the language so long as a clash of variables is avoided), are merely alphabetic variations of the same closed term.
