As you recall from the historical development, Cantor's naive set definition also allowed constructing a "set of all sets". With the axiom of separation, such a set cannot exist.# Corollary: There is no set of all sets

(related to Axiom: Schema of Separation Axioms (Restricted Principle of Comprehension))

There is no such set $X$ which contains the set $\{z\in X\mid z\not\in z\}$ as an element. In other words, a "set of all sets" does not exist.

Proofs: 1

Proofs: 1 2

Thank you to the contributors under CC BY-SA 4.0!




  1. Hoffmann, Dirk W.: "Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011
  2. Ebbinghaus, H.-D.: "Einf├╝hrung in die Mengenlehre", BI Wisschenschaftsverlag, 1994, 3th Edition