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Work over a model of $\sf GCH$. Suppose that $\kappa$ is some regular cardinal, and $U$ is a uniform ultrafilter on $\kappa^+$.

Does $U$ have some canonical extension after forcing with $\operatorname{Add}(\kappa,\kappa)$?

(Alternatively, replace $\kappa^+$ by an arbitrary regular $\lambda>\kappa$.)

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    $\begingroup$ I can't imagine any of the extensions deserving the name "canonical". Each of the subsets $A$ of $\kappa$ that you've added produces a subset of $\kappa^+$, by chopping the ordinal $\kappa^+$ into blocks of length $\kappa$ and putting $A$ on each block. An extension of $U$ has to choose between this set and its complement, and I see no way to make that choice canonical. Perhaps a more rigorous argument would be that no extension of $U$ can be invariant under the automorphisms of the forcing, not even the automorphisms that interchange the generic sets with their complements. $\endgroup$ – Andreas Blass Jul 9 '17 at 15:38
  • $\begingroup$ @Andreas: I feared that would be the answer. Thank you for also addressing the automorphisms issue. If you post this as an answer, I will accept it. $\endgroup$ – Asaf Karagila Jul 9 '17 at 15:41
  • $\begingroup$ Andreas, are there other possibly "slightly more canonical" uniform ultrafilters perhaps, or do all that we can do is use AC to extend the "co-small subsets filter"? $\endgroup$ – Asaf Karagila Jul 9 '17 at 19:47
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At Asaf's suggestion, I'm posting my earlier comment as an answer; I'll also append something about Asaf's subsequent question.

I can't imagine any of the extensions deserving the name "canonical". Each of the subsets $A$ of $\kappa$ that you've added produces a subset of $\kappa^+$, by chopping the ordinal $\kappa^+$ into blocks of length $\kappa$ and putting $A$ on each block. An extension of $U$ has to choose between this set and its complement, and I see no way to make that choice canonical. Perhaps a more rigorous argument would be that no extension of $U$ can be invariant under the automorphisms of the forcing, not even the automorphisms that interchange the generic sets with their complements.

Asaf also asked about "slightly more canonical ultrafilters" versus just using AC to extend the co-small subsets filter. The only improvement I can see is to start with a richer filter then the co-small subsets, for example the club filter. But in the end, one still needs to extend it in what looks to me like a hopelessly non-canonical application of AC.

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  • $\begingroup$ Yeah, the club filter also crossed my mind, as it is the only way I could see the chain conditions and closure working their way into the argument. But, as you say, it must be the case that some new stationary sets are added, and there is no clear choice about them. Thanks... $\endgroup$ – Asaf Karagila Jul 9 '17 at 22:52

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