What does _x_ mean in Set Theory? My mathematical logic textbook defines $\{x \ | \ \text {_} x \text {_} \ \}$, but I'm not sure what the $\text {_} x \text {_}$ means. 
Do the _ just mean 'for any expression involving $x$', or is there something I'm missing?
 A: You can supplement Enderton's explanation with some examples from :


*

*Herbert Enderton, Elements of set theory (1977), page 4 :



The notation used for the set of all objects $x$ such that the condition 
  $\text {__} x \text {__}$,  holds is 
$$\{x \ | \ \text {__} x \text {__} \}.$$
For example: 
  
  
*
  
*$\mathcal PA$ is the set of all objects $x$ such that $x$χ is a subset of $A$. Here "$x$ is a subset of $A$" is the entrance requirement that $x$ must satisfy in order to belong to $\mathcal PA$. We can write 
  
  
  $$\mathcal PA = \{ x \ | \ x \text { is a subset of } A \}$$ 
$$= \{ x \ | x \subseteq A \}.$$

  
*The set $\{ z \ | \ z \ne z \}$ equals $\emptyset$, because the entrance requirement "$z \ne z$" is not satisfied by any object $z$.
  
*The set $\{ n \ | \ n \text { is an even prime number} \}$ is the same as the set $\{ 2 \}$. 

Thus, the set-builder expression $\{x \ | \ \text {__} x \text {__} \}$ needs a condition "$\text {__} x \text {__}$" to be specified; all and only those objects $x$ satisfying the condition will belong to the corresponding set.
A: The symbols in question just mean "any expression involving $x$."
