[FOM] Axiom of Choice/(ultra)filters
Andreas Blass
ablass at umich.edu
Sat Feb 23 19:00:13 EST 2008
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
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