Comments on Existence of a set of all sets
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Existence of a set of all sets
Suppose that we have an axiomatic set theory having the following axiom:
The Axiom Schema of Comprehension: Let $\mathbf{P}(x)$ be a property of $x$. For any set $A$, there is a set $B$ such that $x\in B$ if and only if $x\in A$ and $\mathbf{P}(x)$.
Can a set of all sets exist within such an axiomatic system?
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Let $V$ be a set of all sets. According to the Axiom Schema of Comprehension, we can have the following set:
$$U= \{ x\in V \mid x \not \in x \}.$$
Now, there are two cases:
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$U \in U$, which, according to the definition of the set $U$, implies that $U \not \in U$, which is a contradiction;
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$U \not \in U$, which, according to the definition of the set $U$, implies that $U \in U$, which is a contradiction.
Thus, a set of all sets cannot exist in such an axiomatic system.
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