[FOM] Axiom of Choice/(ultra)filters

[Ilya Tsindlekht]

On Wed, Feb 27, 2008 at 05:43:43PM +0100, pax0 at seznam.cz wrote:
> Johan Belinfante wrote:
[...]
> 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.
Have you considered the case $A_\alpha=\kappa\backslash\{\alpha\}$ ? If
I understood you correct, this is a counterexample to your suggestion
for any non-principal ultrafilter, since the set of all \gamma-th
element will contain at most 2 elements for every \gamma, so J will be
empty.