Understanding the statement $S = \{x : x \in S\}$ In Introduction to analysis by Rosenlicht there is the following set definition: 
$$S = \{x : x \in S\}$$
There is no specific context to this it's just that I find this a little recursive. I would read it "$S$ is a set of $x$ for which every $x$ is an element of $S$". Is that correct?
 A: You are right! This definition is recursive and meaningless. You are also reading it correctly as something like "the set of everything that is an element of S"
The problem here is that there is no definition of set. It is a primary concept. The reason why we don't define "set" is because you would have defined it in terms of other things, which themselves are defined in terms of "lower-level things" and the chain would never end.
So, unless you really want to go deeper into the foundations of mathematics and set theory, for most practical purposes, you can just think of a set as an "arbitrary collection of elements" that is defined ONLY by its elements, and where order does not matter.
For example, if $S:=\{1,2,3\}, T:=\{2,3,1\}$ and $U:=\{x \in \mathbb{Z} : 0<x<4\}$, then $S, T$ and $U$ are the same set.
A: The so-called Set-builder notation is used to "specify" a set. 
Its syntax is 

$S = \{ x \mid \varphi(x) \}$

where formula $\varphi(x)$ expresses a property (a condition).
The meaning is : 

"the set $S$ is made of all and only those object that satisfy condition $\varphi(x)$".

As a way to define a new set $S$, it makes little sense to use $x ∈ S$ as "specifying condition".
But obviously, if we are not defining $S$ but we already know it, then we have trivially that $S = \{ x \mid x∈S \}$.
