A $P$-point is an ultrafilter $\scr U$on $\omega$ such that for every function $f:\omega\to\omega$ there is $x\in {\scr U}$ such that the restriction $f|_x$ is either constant, or injective.

A $Q$-point is an ultrafilter $\scr U$on $\omega$ such that for every function $f:\omega\to\omega$ with the property that $f^{-1}(\{m\})$ is finite for each $m\in \omega$, there is $x\in {\scr U}$ such that the restriction $f|_x$ is injective.

$P$-points need not exist, and $Q$-points need not exist.

Question. Is it possible that neither $P$- nor $Q$-points exist?

A $P$-point is an ultrafilter $\scr U$on $\omega$ such that for every function $f:\omega\to\omega$ there is $x\in {\scr U}$ such that the restriction $f|_x$ is either constant, or finite-to-one.

A $Q$-point is an ultrafilter $\scr U$on $\omega$ such that for every function $f:\omega\to\omega$ with the property that $f^{-1}(\{m\})$ is finite for each $m\in \omega$, there is $x\in {\scr U}$ such that the restriction $f|_x$ is injective.

$P$-points need not exist, and $Q$-points need not exist.

Question. Is it possible that neither $P$- nor $Q$-points exist?