[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

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