Editing Burali-Forti paradox (section)

==Stated more generally==

The version of the paradox above is anachronistic, because it presupposes the definition of the ordinals due to [[John von Neumann]], under which each ordinal is the set of all preceding ordinals, which was not known at the time the paradox was framed by Burali-Forti.
Here is an account with fewer presuppositions:  suppose that we associate with each [[well-ordering]]
an object called its [[order type]] in an unspecified way (the order types are the ordinal numbers).  The order types (ordinal numbers) themselves are well-ordered in a natural way,
and this well-ordering must have an order type <math>\Omega</math>.  It is easily shown in
[[naive set theory|naïve set theory]] (and remains true in [[ZFC]] but not in [[New Foundations]]) that the order
type of all ordinal numbers less than a fixed <math>\alpha</math> is <math>\alpha</math> itself.
So the order
type of all ordinal numbers less than  <math>\Omega</math> is <math>\Omega</math> itself.  But
this means that <math>\Omega</math>, being the order type of a proper initial segment of the ordinals,  is strictly less than the order type of all the ordinals,
but the latter is <math>\Omega</math> itself by definition.  This is a contradiction.

If we use the von Neumann definition, under which each ordinal is identified as the set of all preceding ordinals, the paradox is unavoidable:  the offending proposition that the order type of all ordinal numbers less than a fixed <math>\alpha</math> is <math>\alpha</math> itself must be true.  The collection of von Neumann ordinals, like the collection in the [[Russell paradox]], cannot be a set in any set theory with classical logic.  But the collection of order types in New Foundations (defined as equivalence classes of well-orderings under similarity) is actually a set, and the paradox is avoided because the order type of the ordinals less than <math>\Omega</math>
turns out not to be <math>\Omega</math>.