# Is there a model of set theory in which $\mathfrak p< \mathfrak b < \mathfrak q$?

Is there a model of set theory in which $\mathfrak p< \mathfrak b < \mathfrak q$?

Here $\mathfrak p$, $\mathfrak b$, $\mathfrak q$ are small uncountable cardinals:

• $\mathfrak p$ is the smallest cardinality of any family $\mathcal F \subseteq [\omega]^\omega$, which has the strong finite intersection property, but does not have a pseudo intersection;
• $\mathfrak b$ is the bounding number;
• $\mathfrak q = \min\{\kappa :\text{ no subset }X \subseteq \mathbb R\text{ of cardinality }|X| \ge κ\text{ is a Q-set}\}$.

• $\min\{\mathfrak b, \mathfrak q \}\in \{\mathfrak p,\mathfrak q \}$ ?
• Probably you should clarify a bit what you mean by p, b and q. Although a reasonable guess would be that you mean small uncountable cardinals. I.e., $\mathfrak p$ is the smallest cardinality of system which has s.f.i.p. but now infinite pseudointersection. (Do some people call this power number?) And $\mathfrak b$ is bounding number. I am not sure about $\mathfrak q$. – Martin Sleziak May 26 '18 at 18:42
• p is the smallest cardinality of system which has s.f.i.p. but now infinite pseudointersection. b is bounding number. q = min{κ : no subset X ⊂ R of cardinality |X| ≥ κ is a Q-set}. – Alexander Osipov May 26 '18 at 18:53
• @Alexander Could you put it in the body text? – Harry Gindi May 26 '18 at 19:01
• @AlexanderOsipov I have added at least this basic information into the text. If you have something more that could be useful for readers of your question, please do edit it further. – Martin Sleziak May 26 '18 at 19:03
• I have seen $\mathfrak{q}$ be defined as the minimal size of a set of reals which is not a Q-set. Is this definition equivalent to the one given in the question? – Ramiro de la Vega May 27 '18 at 0:43