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Ordinals fixed by T are called <strong>cantorian</strong> ordinals, and ordinals which
dominate only cantorian ordinals (which are easily shown to be cantorian themselves) are said to be <strong>strongly cantorian</strong>. The motivation for these definitions will be clearer in the next section. It is important to note that there can be no set of cantorian ordinals or set of
strongly cantorian ordinals.
It is possible to reason about von Neumann ordinals in [[NFU]]. Recall that a von Neumann ordinal
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This is quite a strong condition in the [[NFU]] context, since the membership relation involves
a difference of type. A von Neumann ordinal A is not an ordinal in the sense of [[NFU]], but <math>\in\lceil A</math> belongs to an ordinal <math>\alpha</math> which may be termed
the order type of (membership on) A. It is easy to show that the order type of a von Neumann
ordinal A is cantorian: for any well-ordering W of order type <math>\alpha</math>, the induced
well-ordering of initial segments of W by inclusion has order type <math>T(\alpha)</math> (it is one type higher, thus the application of T): but the well-orderings of a von Neumann ordinal A
by membership and inclusion are clearly the same because the two relations are actually the
same relation, so the order type of A is fixed under T. Moreover, the same argument applies
to any smaller ordinal Iwhich will be the order type of an initial segment of A, also a von
Neumann ordinal) so the order type of any von Neumann ordinal is strongly cantorian!
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".
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