a set. The axioms of replacement, global choice, and union (with the other axioms of NBG) imply the axiom of limitation of size. Therefore, this axiom is equivalent to the combination of replacement, global choice, and union in NBG or Morse–Kelley set theory. These set theories only substituted the axiom of replacement and a form of the axiom of choice for the axiom of limitation of size because von Neumann's axiom system contains the axiom of union. Lévy's proof that this axiom is redundant came many years later. The axioms of NBG with the axiom of global choice replaced by the usual axiom of choice do not imply the axiom of limitation of size. In 1964, William B. Easton used forcing to build a model of NBG with global choice replaced by the axiom of choice. In Easton's model, V cannot be linearly ordered, so it cannot be well-ordered. Therefore, the axiom of limitation of size fails in this model. Ord is an example of a proper class that cannot be mapped onto V because (as proved above) if there is a function mapping Ord onto V, then V can be well-ordered. The axioms of NBG with the axiom of replacement replaced by the weaker axiom of separation do not imply the axiom of limitation of size. Define as the -th infinite initial ordinal, which is also the cardinal ; numbering starts at , so In 1939, Gödel pointed out that Lωω, a subset of the constructible universe, is a model of ZFC with replacement replaced by separation. To expand it into a model of NBG with replacement replaced by separation, let its classes be the sets of Lωω+1, which are the constructible subsets of Lωω. This model satisfies NBG's class existence axioms because restricting the set variables of these axioms to Lωω produces instances of the axiom of separation, which holds in L. It satisfies the axiom of global choice because there is a function belonging to Lωω+1 that maps ωω onto Lωω, which implies that Lωω is well-ordered. The axiom of limitation of size fails because the proper class {ωn : n ∈ ω} has cardinality , so it cannot be mapped onto Lωω, which has cardinality . In a 1923 letter to Zermelo, von Neumann stated the first version of his axiom: A class is a proper class if and only if there is a one-to-one correspondence