Proof
(related to Corollary: There is no set of all sets)
- We have seen in the above explanation how the axiom of separation avoids the Russell's Paradox.
- In particular, we have seen that for every set $X$ the set $\{z\in X\mid z\not\in z\}$ is well-defined (exists).
- Therefore, if a "set of all sets" existed, it would have to contain $\{z\in X\mid z\not\in z\}$ as an element.
- But for all $X$ we have $\{z\in X\mid z\not\in z\}\not\in X.$
- Therefore, a "set of all sets" does not exist.
∎
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References
Bibliography
- Hoffmann, Dirk W.: "Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011
- Ebbinghaus, H.-D.: "Einführung in die Mengenlehre", BI Wisschenschaftsverlag, 1994, 3th Edition
