There is no set to which every set belongs I am reading Elements of Set Theory by Enderton H.B.
This is probably a simple misunderstanding, but I have become confused by the proof of the first theorem in the book.
The theorem is as follows: "There is no set to which every set belongs"
The proof starts by saying:
"Let A be a set; we will construct a set not belonging to A". Let
$$
B=\{x\in A:x\notin x\}
$$
This goes on to show that $B\notin A$.
I understand the rest of the proof, but feel that this initial construction is not well founded.
Is the initial set A that was chosen the set to which every set belongs? If not, how can we say $x\in A$. And if this is the case then it seems to make more sense to me, but nowhere has that been specified.
 A: Why should $B$ not be well founded (or rather: properly defined)? The comprehension used to define $B$, i.e. to select elements from a given set by a specific property, is the main method to describe a set. Here, $B$ is the subset of $A$ that consists of precisely those elements o$x$ of $A$ for which the deus-ex-machina property $x\notin x$ holds. (Btw. "(well) founded" has a very specific meaning in set thoery that is not the same as "defined")
The initial set $A$ was not the set to which every set belongs - because there is no such set (as theproof is about to show). The proof starts "Let $A$ be a set." Hence $A$ is just any set. If there were a universal set, then you might pick $A$ to be the universal set. However, the proof does not assume so. Instead it exhibits a specific set $B$ that is not element of $A$ and hence shows that the arbitrary set $A$ is not universal. In other words: There is no universal set.
You could rewrite the proof such that $A$ is universal, i.e do a proof by contradiction: "Assume otherwise and let $A$ be a set such that every set is an element of $A$. ... yada yada ... Hence $B\notin A$, contradicting the assumption that $A$ is universal. We conclude that the assumption was false.  In other words: There is no universal set."
A more direct proof, such as in the text, is often more elegant.
