[FOM] Axiom of Choice/(ultra)filters

pax0@seznam.cz pax0 at seznam.cz
Wed Feb 27 11:43:43 EST 2008

Johan Belinfante wrote:

> What exactly do you mean by filter?  Filter for which partial orders?

To make sense of a notion of selector, I use filter for subsets of an arbitrary set, ordered by inclusion.
I add one more question on filters:
Let F be a filter on cardinal \kappa and let \kappa members of F A_\alpha, \alpha < \kappa be given.
Let J be the set of all ordinals \gamma < \kappa such that \gamma-th elements (of A_\alpha together) are in F . 
Can we demand that one of the following conditions holds for all sequences A_alpha?
(1) card(J)=\kappa,
(2) J is cofinal in \kappa,
(3) J is in F.

Are there any equivalent properties for \kappa to have filter satisfying (1), (2) or (3) respectively?

Thank you,JP

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