There is no set of all sets

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Statement

We claim there is no set of all sets Claim:

• $\forall X\exists Y[Y\notin X]$
  • For any set $X$ there exists a set $Y$ such that $Y\notin X$

Significance

If we take $X$ to be the "set of all sets" we'd reach Russell's paradox, this statement shows half of it. The remainder of the paradox shows that $Y\notin X$ is absurd too.

Proof

• Let $X$ be given
  • Define $Y:\eq\{x\in X\ \vert\ x\notin x\}$ - which exists by the axiom schema of separation, accordingly $Y\subseteq X$
    • Suppose that $Y\in X$
      • We now have two cases (as $Y\in Y$ is conceivable (as is $Y\notin Y$) as $Y$ is some subset of $X$):
        1. $Y\in Y$
           • But $\forall A\in Y[A\notin A]$ is the defining property of $Y$
             • So then $Y\notin Y$ (the case where $A:\eq Y$) - a contradiction! So we cannot have this!
        2. $Y\notin Y$
           • As $Y\in X$ by hypothesis, if $Y\notin Y$ then $Y\in Y$ by definition of what $Y$ contains from $X!$
             • A contradiction again! So we cannot have this!
      • Thus $Y\notin X$

Note: this is half of Russell's paradox, the other half is showing that $Y\notin X$ is absurd too!.