 is an assertion which makes sense in TST and is in fact true in TST and so in NFU. We thus find that the order type of the restriction of the natural order on the ordinals to the ordinals less than \(\Omega\) is \(T^2 (\Omega)\), whence we find that \(T^2 (\Omega)\) (as the order type of a proper initial segment of the ordinals) is strictly less than \(\Omega\) (which is the order type of all the ordinals). Once again, the fact that the singleton map is not a function eliminates the “intuitively obvious” similarity between these orders. This also shows that \(T\) is not a function. \(T\) is an order endomorphism of the ordinals, though, whence we have \(\Omega \gt T^2 (\Omega) \gt T^4 (\Omega)\ldots\), which may be vaguely disturbing, though this “sequence” is not a set. A perhaps useful comment is that in the models of NFU described above, the action of \(T\) on ordinals exactly parallels the action of \(j\) on order types of well-orderings \((j\) does not send NFU ordinals to ordinals, exactly, so this needs to be phrased carefully): the “descending sequence” already has an analogue in the sequence \(\alpha \gt j(\alpha) \gt j^2 (\alpha)\ldots\) in the original nonstandard model. Some have asserted that this phenomenon (that the ordinals in any model of NFU are not externally well-ordered) can be phrased as “NFU has no standard model”. We reserve judgement on this—we do note that the theorem “the ordinals in any (set!) model of NFU are not well-ordered” is a theorem of NFU itself; note that NFU does not see the universe as a model of NFU (even though it is a set) because the membership relation is not a set relation (if it were, the singleton map certainly would be).