[FOM] Axiom of Choice/(ultra)filters

[Andreas Blass]

Jan Pax asked:

> [W]hat is the strength of the following two theories:
>
> (1)  ZF+{on every filter, there is a selector},
>
> (2)  ZF+{on every ultrafilter, there is a selector} .
>
> More precisely, are they strictly weaker then ZFC?

The first is equivalent to ZFC; the second is strictly weaker.

For the first, assume (1) and suppose, toward a contradiction, that X  
is a set that admits no well-ordering.  Then the well-orderable  
subsets of X form a proper ideal of subsets of X, since the union of  
two well-orderable sets is well-orderable.  Applying (1) to the  
filter of complements of these sets, we obtain a selector S that  
chooses, from the complement A of any well-orderable subset of X,  
some element S(A) in A.  Using S, we can define, by transfinite  
induction, a one-to-one function F from the ordinals into A, by setting
	F(alpha) = S(X - {F(beta): beta < alpha}).
Since the ordinals form a proper class while X is a set, this is a  
contradiction.

For the second, I use the result in my paper "A model without  
ultrafilters" (Bull. Acad. Polon. Sci. 25 (1977) 329-331) that there  
is a model of ZF in which all ultrafilters are principal.  Such a  
model trivially satisfies (2) but not AC.

Andreas Blass