I was interested in the Grigorieff forcing (you can read the definition here: [How "much" does (Grigorieff) forcing destroy an ultrafilter?](https://mathoverflow.net/questions/62981/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.



Grigorieff forcing and destruction of ultrafilters
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.




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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.


