A non-principal ultra filter containing the even numbers, need hint now. I posted a question about an exercise asking to prove that there exists a non-principal ultra filter on N containing the set of even numbers.  My original post asked about a possible answer.  It was pointed out that it was wrong, and I saw that I had a lot more to think through.  I'm pretty stuck, and would appreciate a hint.
I tried various constructions using rules of including even numbers, but I can always find a counter example where some element and its complement are both in or out.  Or, I wind up with a singleton via finite intersections.
Can anyone give me a hint or suggestion?  By the way I don't know how to reference my original question in this one.
 A: Given that there is a non-principle u.f. $A$ on N, if it contains the evens, you are done. 
Otherwise, $A$ contains the odds and you can take it image $B$ under a bijection $f$ on N which exchanges evens and odds-- for example, $f(2n) = 2n+1, f(2n+1)=2n$. 
[ i.e. $B$ = { $f(X)$  | $X$ is in $A$ } ].
A: You can't constructively exhibit such an ultrafilter, so you will have to appeal to Zorn's lemma. Here's a hint that should get you going. Prove the following lemma, then use the fact that every filter can be extended to an ultrafilter.

Lemma. If $\cal F$ is a filter on $X$, and $A\subseteq X$ such that for all $B\in\cal F$, $A\cap B\neq\varnothing$, then there is a filter $\cal F'$ such that $\mathcal F\cup\{A\}\subseteq\cal F'$.

Now find a filter which cannot be extended to a principal ultrafilter, such that the even numbers are a member of that filter.
A: You cannot find an explicit example, because ultrafilters need not exist with the axiom of choice (bit of a short explanation, Asaf can explain this better...). So nothing like a simple formula that describes the sets in the ultrafilter (an ultrafilter is a non-measurable set in the power set...).
So prove, using Zorn's lemma, that every filter can be extended to an ultrafilter. You can then start with the filter of all supersets of the even numbers, and also containing all co-finite sets.
