[FOM] free ultrafilters
Andreas Blass
ablass at umich.edu
Fri Oct 13 09:25:37 EDT 2006
Bill Taylor asked:
> Is it possible/easy to describe in some way, how to have two non-
> isomorphic
> free ultrafilters on the integers, where isomorphism is defined
> with respect to permutations on the integers.
A lot depends here on the meaning of "have". If you want a
*definition* of two non-isomorphic free ultrafilters, then the answer
is negative. In fact, you can't expect to get a definition for even
one free ultrafilter on the integers. It is consistent, relative to
ZFC, that no free ultrafilter on the integers is definable, even with
real and ordinal numbers allowed as parameters in the definition.
(Of course definitions become possible if one assumes G"odel's axiom
of constructibility, which provides a definable well-ordering of the
universe and thus allows one to define things, like free
ultrafilters, that usually come from the axiom of choice.)
If, on the other hand, you just want a proof that two non-isomorphic
free ultrafilters exist, then this can be done by a counting
argument. The number of permutations of the integers is the
cardinality c of the continuum, so any one free ultrafilter is
isomorphic to at most c others (in fact, exactly c, but that's not
needed here). But it is known that there are 2^c free ultrafilters
on the integers. So they can't all be in one isomorphism class; in
fact there have to be 2^c isomorphism classes.
Sincerely yours,
Andreas Blass
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