[FOM] Axiom of Choice/(ultra)filters

[Johan Belinfante]

pax0 at seznam.cz wrote:
> I have a question regarding the Axiom of Choice,
> what 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?
>
> Here, by a selector on a set X I mean a function with the domain all nonempty sets in X and
> with f(x) \in x for every member x in its domain.
> Then ZFC is equivalent to ZF+{on every set, there is a selector}.
>
> Thank you
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What exactly do you mean by filter?  Filter for which partial orders?