[FOM] free ultrafilters
Andreas Blass
ablass at umich.edu
Sat Oct 14 17:33:40 EDT 2006
Thomas Forster wrote:
> I think what Bill was after (his query arose in a conversation we were
> having) was the cutest/simplest (pair of) constructions of two
> nonprincipal ufs over N - using AC of course.
The simplest "construction" that comes to mind is to build the two
ultrafilters simultaneously, by a transfinite induction of length c
(the cardinal of the continuum, regarded as an initial ordinal),
preventing, at each step, one permutation from being an isomorphism.
One uses a family of independent sets to make sure the construction
doesn't end before c steps, so there's enough "time" to handle all
the permutations. This argument is due to Kenneth Kunen,
"Ultrafilters and independent sets," Trans. A. M. S. 172 (1972) pp.
299-306. Kunen actually constructs two free ultrafilters on the
integers with the property that no function (permutation or not) maps
either ultrafilter to the other.
Andreas Blass
More information about the FOM
mailing list