Axiom of limitation of size

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In class theories, the axiom of limitation of size says that for any class C, C is a proper class (a class which is not a set (an element of other classes)) if and only if V (the class of all sets) can be mapped one-to-one into C.

$\forall C [\lnot \exist W (C \in W) \iff \exist F ( \forall x [\exist W (x \in W) \Rightarrow \exist s (s \in C \and \langle x, s \rangle \in F)] \and$

Failed to parse (Cannot write to or create math output directory): \forall x \forall y \forall s [(\langle x, s \rangle \in F \and \langle y, s \rangle \in F) \Rightarrow x = y])].

This axiom is due to John von Neumann. It implies the axiom schema of specification, axiom schema of replacement, and axiom of global choice at one stroke. The axiom of limitation of size implies the axiom of global choice because the class of ordinals is not a set, so there is an injection from the universe to the ordinals. Thus the universe of sets is well-ordered.

Although together the axiom schema of replacement and the axiom of global choice (with the other axioms of Morse–Kelley set theory) imply this axiom, they are each at least as complicated as the axiom of limitation of size and no more intuitive (once you understand this axiom). So using this axiom instead of them is a net improvement.