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Peter Krautzberger
Peter Krautzberger
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Another way to get a qQ-filter that is not an ultrafilter is to usedestroy a Q-point using an $\omega^\omega$-bounding forcing, e.g., Grigorieff forcing with a selective ultrafilter.

Grigorieff forcing destroys the selective ultrafilter, but because Grigorieff forcing with a P-filter is $\omega^\omega$-bounding, the (formerly selective) filter will remain a qQ-filter.

Another way to get a q-filter that is not an ultrafilter is to use Grigorieff forcing with a selective ultrafilter.

Grigorieff forcing destroys the selective ultrafilter, but because Grigorieff forcing with a P-filter is $\omega^\omega$-bounding, the (formerly selective) filter will remain a q-filter.

Another way to get a Q-filter that is not an ultrafilter is to destroy a Q-point using an $\omega^\omega$-bounding forcing, e.g., Grigorieff forcing with a selective ultrafilter.

Grigorieff forcing destroys the selective ultrafilter, but because Grigorieff forcing with a P-filter is $\omega^\omega$-bounding, the (formerly selective) filter will remain a Q-filter.

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Peter Krautzberger
Peter Krautzberger
  • 655
  • 7
  • 18

Another way to get a q-filter that is not an ultrafilter is to use Grigorieff forcing with a selective ultrafilter.

Grigorieff forcing destroys the selective ultrafilter, but because Grigorieff forcing with a P-filter is $\omega^\omega$-bounding, the (formerly selective) filter will remain a q-filter.