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I was interested in the Grigorieff forcing (you can read the definition here: How "much" does (Grigorieff) forcing destroy an ultrafilter?)
I couldn't prove that it destroys ultrafilters, and I couldn't find the proof anywhere.
I would be happy if you could write the proof or give me a reference.
Also what other forcing notions deliberately destroys ultrafilters?

Thanks,
Danny.

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Let $g$ be the generic subset of $\omega$ added by the Grigorieff forcing, $G(F)$, where $F$ was a free ultrafilter.

It is easy to see that for every $A\in F$, $g\cap A$ is non-empty, since if $f$ is any condition in $G(F)$, then there is some $n\in A\setminus\operatorname{dom} f$ and we can simply take $f\cup\{(n,1)\}$ as a condition. But it's also easy to see that by the same argument $\omega\setminus g$ meets every $A\in F$.

In other words, the filter generated from $F$ in $V[g]$ cannot decide whether $g$ is in or out. So it is not an ultrafilter.

More generally, every forcing whose generic and its complement both meet all sets in an ultrafilter will destroy it.

So Cohen reals will do the job real nice.

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