Axiom of Choice/(ultra)filters

[pax0 at seznam.cz]

Dear FOMers,

many years ago I asked here a question (copied below) about selectors on all
(ultra)filters.

I got this answer:




>(1) as strong as ZFC, but (2) weaker.




When I went through the proof recently, I realized that in (1) there is a 
kind of unwanted triviality exploited in the proof

(obtaining AC from (1)). So I would now modify it to

(1 ") ZF+{on every NON-PRINCIPAL filter, there is a selector}


Does this modified (1 ") still imply full (AC) ?

What is its relationship with (2) ?


---End of my question---




Thank you, Jan Pax




"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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