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which is equinumerous with the field of W and membership on which is isomorphic to the strict
well-ordering associated with W. (the equinumerousness condition distinguishes between well-orderings with fields of size 0 and 1, whose associated strict well-orderings are indistinguishable).
In [[ZFC]] there cannot be a set of all ordinals. In fact, the von Neumann ordinals are an
inconsistent totality in any set theory: it can be shown with modest set theoretical assumptions
that every element of a von Neumann ordinal is a von Neumann ordinal and the von Neumann
ordinals are strictly well-ordered by membership. It follows that the class of von Neumann
ordinals would be a von Neumann ordinal if it were a set: but it would then be an element
of itself, which contradicts that fact that membership is a strict well-ordering of the
von Neumann ordinals.
In [[NFU]], the collection of all ordinals is a set by stratified comprehension. The Burali-Forti paradox is evaded in an unexpected way. There is a natural order on the ordinals
defined by <math>\alpha\leq \beta</math> iff some (and so any) <math>W_1 \in \alpha</math> is similar
to an initial segment of some (and so any) <math>W_2\in \beta</math>. Further, it can be
shown that this natural order is a well-ordering of the ordinals and so must have an order
type <math>\Omega</math>. It would seem that the order type of the ordinals less than
<math>\Omega</math> with the natural order would be <math>\Omega</math>, contradicting
the fact that <math>\Omega</math> is the order type of the entire natural order on the ordinals
(and so not of any of its proper initial segments). But this relies on our intuition (correct
in [[ZFC]]) that the order type of the natural order on the ordinals less than <math>\alpha</math>
is <math>\alpha</math> for any ordinal <math>\alpha</math>. This assertion is unstratified,
because the type of the second <math>\alpha</math> is four higher than the type of the first
(two higher if a type level pair is used). The assertion which is true and provable in
[[NFU]] is that the order type of the natural order on the ordinals less than <math>\alpha</math>
is <math>T^4(\alpha)</math> for any ordinal <math>\alpha</math>, where <math>T(\alpha)</math> is the
order type of <math>W^{\iota}=\{(\{x\},\{y\})\mid xWy\}</math> for any <math>W \in \alpha</math> (it is easy to show that this does not depend on the choice of W; note that T raises type by one). Thus the order type of the ordinals less than
<math>\Omega</math> with the natural order is <math>T^4(\Omega)</math>, and <math>T^4(\Omega)<\Omega</math>. All uses of <math>T^4</math> here can be replaced with <math>T^2</math> if a type-level pair is used.
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