[FOM] A new definition of Cardinality.

[T.Forster@dpmms.cam.ac.uk]

There are various problems with approaches along these lines. One is that 
the construction of Scott's-trick cardinals seems to need replacement. 
Another is that - if one wants the implementation of cardinals to succeed 
even when foundation fails - then one runs up against the fact that it is 
not a theorem of ZF that every set is the same size as a wellfounded set, 
so wellfounded objects cannot supply cardinals for everything in the 
universe. To see this, start with a model of ZFC; add infinitely many Quine 
atoms (x = {x}) by a Rieger-Bernays permutation, so you have a model of ZFC 
+ lots of Quine atoms; then do an FM construction to make AC fail in the 
cone above the Quine atoms. Choice still holds in the wellfounded part of 
the universe (which is iso to what you started with, in fact). So there are 
illfounded sets which are not the same as any wellfounded set.

The principle that every set is the same size as a wellfounded set 
(isolated first by Jean Coret) has nice logical features - but i digress!