Axiom of limitation of size

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

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

$\forall x \forall y \forall s [(\langle s, x \rangle \in F \and \langle s, y \rangle \in F) \Rightarrow x = y])].$

This axiom is due to John von Neumann. It implies the axiom schema of specification, axiom schema of replacement, axiom of global choice, and even, as noticed later by Azriel Levy, axiom of union 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 a surjection from the ordinals to the universe, thus an injection from the universe to the ordinals, that is, the universe of sets is well-ordered.

Together the axiom of replacement and the axiom of global choice (with the other axioms of von Neumann–Bernays–Gödel set theory) imply this axiom. This axiom can then replace replacement, global choice, specification and union in von Neumann–Bernays–Gödel or Morse–Kelley set theory.

It can be shown that a class is a proper class if and only if it is equinumerous to V, but von Neumann's axiom does not capture all of the "limitation of size doctrine", because the axiom of power set is not a consequence of it. Later expositions of class theories (Bernays, Gödel, Kelley, ...) generally use replacement and a form of axiom of choice rather than axiom of limitation of size.

See also

• Axiom of global choice
• Limitation of size
• Von Neumann–Bernays–Gödel set theory
• Morse–Kelley set theory

References

• Abraham Fraenkel, Yehoshua Bar-Hillel, and Azriel Levy, 1973 (1958). Foundations of Set Theory. North Holland.