Page "Cardinal assignment" Paragraph 6

Formally, assuming the axiom of choice, cardinality of a set X is the least ordinal α such that there is a bijection between X and α.

This definition is known as the von Neumann cardinal assignment.

If the axiom of choice is not assumed we need to do something different.

The oldest definition of the cardinality of a set X ( implicit in Cantor and explicit in Frege and Principia Mathematica ) is as the set of all sets which are equinumerous with X: this does not work in ZFC or other related systems of axiomatic set theory because this collection is too large to be a set, but it does work in type theory and in New Foundations and related systems.

However, if we restrict from this class to those equinumerous with X that have the least rank, then it will work ( this is a trick due to Dana Scott: it works because the collection of objects with any given rank is a set ).