Editing Cardinal assignment (section)

== Cardinal assignment without the axiom of choice ==

Formally, assuming the axiom of choice, the 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 that 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 (set theory)|class]] to those equinumerous with ''X'' that have the least [[rank (set theory)|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; see [[Scott's trick]]).