# Axiom of limitation of size

In class theories, the

**axiom of limitation of size**says that for any class "C", "C" is aproper 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".:$forall\; C\; [lnot\; exist\; W\; (C\; in\; W)\; iff\; exist\; F\; (\; forall\; x\; [exist\; W\; (x\; in\; W)\; Rightarrow\; exist\; s\; (s\; in\; C\; and\; langle\; x,\; s\; angle\; in\; F)]\; and$::$forall\; x\; forall\; y\; forall\; s\; [(langle\; x,\; s\; angle\; in\; F\; and\; langle\; y,\; s\; angle\; in\; F)\; Rightarrow\; x\; =\; y]\; )]\; .$

This axiom is due to

John von Neumann . It implies theaxiom schema of specification ,axiom schema of replacement , andaxiom 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 iswell-order ed.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.

**ee also***

Axiom of global choice

*Limitation of size

*Von Neumann–Bernays–Gödel set theory

*Morse–Kelley set theory

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