There may be exactly $n$ $Q$-points

TL;DR

This paper extends previous results to show that for any finite number n, it is consistent that exactly n Q-points exist up to isomorphism, using advanced forcing techniques.

Contribution

It generalizes the existence and uniqueness results of Q-points to any finite n and introduces a new forcing iteration method for n=2.

Findings

Consistency of exactly n Q-points for any finite n

For n=2, a new forcing iteration achieves the result

Extends the main theorem of arXiv:2505.17960

Abstract

We generalize the main result of arXiv:2505.17960 and show the consistency of the statement ``There are exactly $n$ $Q$-points up to isomorphism" for any finite $n$. Furthermore, we show that the above statement for $n=2$ can alternatively be obtained by a length-$\omega_2$ countable support iteration of Matet-Mathias forcing restricted to a Matet-adequate family.