[FOM] A new definition of Cardinality.
T.Forster@dpmms.cam.ac.uk
T.Forster at dpmms.cam.ac.uk
Sun Nov 29 03:43:39 EST 2009
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!
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