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The only von Neumann ordinals which can be shown to exist in NFU without additional assumptions are the concrete finite ones. However, the application of a permutation method can convert any model of NFU to a model in which every strongly cantorian ordinal is the order type of a von Neumann ordinal. This suggests that the concept "strongly cantorian ordinal of NFU" might be a better
analogue to "ordinal of ZFC" than is the apparent analogue "ordinal of NFU".
== Cardinal numbers ==
Cardinal numbers are defined in [[NFU]] in a way which generalizes the definition of natural
number: for any set A, <math>|A| \equiv_{def} \{B \mid B \sim A\}</math>.
In [[ZFC]], these equivalence classes are too large as usual. Scott's trick could be used
(and indeed is used in [[ZF]] without Choice), but we usually define <math>|A|</math> as the
smallest order type (here a von Neumann ordinal) of a well-ordering of A (that every set can be well-ordered follows from
the Axiom of Choice in the usual way in both theories).
The natural order on cardinal numbers is seen to be a well-ordering: that it is reflexive,
antisymmetric (on abstract cardinals, which are now available) and transitive has been shown
above. That it is a linear order follows from the Axiom of Choice: well-order two sets and an
initial segment of one well-ordering will be isomorphic to the other, so one set will have cardinality smaller than that of the other. That it is a well-ordering follows from the Axiom of Choice in a similar way.
With each infinite cardinal, many order types are associated for the usual reasons (in either
set theory).
The operations of cardinal arithmetic are defined in a set-theoretically motivated way in
both theories. <math>|A| + |B| = \{C \cup D \mid C \sim A \wedge D \sim B \wedge C \cap D = \emptyset\}</math>. One would like to define <math>|A|\cdot|B|</math> as <math>|A \times B|</math>,
and one does this in [[ZFC]], but there is an obstruction in [[NFU]] when using the Kuratowski pair:
one defines <math>|A|\cdot|B|</math> as <math>T^{-2}(|A \times B|)</math> because of the
type displacement of 2 between the pair and its projections, which implies a type displacement
of two between a cartesian product and its factors. It is straghtforward to prove that the product
always exists (but requires attention because the inverse of T is not total).
Now the usual theorems of cardinal arithmetic with the axiom of choice can be proved, including
<math>\kappa \cdot \kappa = \kappa</math>. From the case <math>|V|\cdot |V| = |V|</math>
we can derive the existence of a type level ordered pair.
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