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Suppose $\kappa$ is a measurable cardinal and let $\mathcal{F}\subset\wp(\kappa)$ be a $\kappa$-complete non-principal filter. Can we extend $\mathcal{F}$ to a $\kappa$-complete ultrafilter?

My motivation comes from a problem I have just encountered. I need a $\kappa$-complete ultrafilter whereas the best I can do is to construct only some filter.

Of course, the standard Zorn argument does not work here by a simple meta-argument. Take the filter on $\omega_1$ consisting of sets with countable complement. It is $\omega_1$-complete but there is no $\omega_1$-complete ultrafilter on $\omega_1$, since this cardinal is not measurable (in ZFC).

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# Extending complete filters

Suppose $\kappa$ is a measurable cardinal and let $\mathcal{F}\subset\wp(\kappa)$ be a $\kappa$-complete non-principal filter. Can we extend $\mathcal{F}$ to a $\kappa$-complete ultrafilter?

My motivation comes from a problem I have just encountered. I need a $\kappa$-complete ultrafilter whereas the best I can do is to construct only some filter.