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I am reading Enderton's Set Theorey, in which he showed proof for a theorem: There is no set to which every set belongs. In the proof, he wrote:

Let A be a set; we will construct a set not belonging to A. Let $B=\{x\in A|x\not \in x\}$.

I have no trouble understanding the proof. My question is: Beside the fact that $A$ is a set whose members are $x$, what is $A$?

$A$ is a set of what?

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    What exactly is meant in $B=\{x\in A|x\not \in x\}$ I don't understand how this means that $B$ is not a subset of $A$? Can't we simply write $B = \{x|x \not \in A\}$?2011-06-03
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    @Tretwick: $B$ **is** a subset of $A$. But $B$ does not **belong** to $A$ - that is, $B\notin A$. The definition $B=\{x\in A|x\not \in x\}$ is correct.2011-06-03
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    @ Zev Chonoles:Thanks,I was confused with "$x\notin x$".Is [this](http://www.math.ucla.edu/~hbe/ency.pdf) same book the OP is talking about?2011-06-03
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    @Zec Chonoles:B is a subset of A but B does not belong to A..how this is possible? :/2011-06-03
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    @Tretwick: "$B$ belongs to $A$" is one way of saying $B\in A$. The distinction is between $B\subset A$ or $B\in A$. For example, $1\in\mathbb{R}$, but $1\not\subset\mathbb{R}$; and $\{1\}\notin\mathbb{R}$, but $\{1\}\subset\mathbb{R}$.2011-06-03
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    Something which perhaps help us understand why no set can be big enough to contain the whole universe, that is, what exactly is it that is generating all these sets we can't fit into any A? (We have exhibited a B, but don't really have an idea of what other obstructions there might be.) Have a think about why we can't fit the ordinals in any set: because of Mostowski collapsing and Hartogs' lemma.2011-06-03
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    If one assumes the axiom of foundation $B=\emptyset$, and if $A$ is the powerset of the empty set, then $B\in A$...2011-06-03

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