Models in Which $\mathfrak {p}=\mathfrak {c}$

Abstract

In this chapter we shall consider models of ZFC in which $\mathfrak {p}=\mathfrak {c}$. Since $\omega_{1}\le \mathfrak {p}$ (by Theorem 8.1) and $\mathfrak {p}\le \mathfrak {c}$, we have $\mathfrak {p}=\mathfrak {c}$ in all models in which $\mathfrak {c}=\omega_{1}$, but of course, these are not the models we are interested in.

By Theorem 13.6 we know that MA(σ-centred) (and therefore MA) implies $\mathfrak {p}=\mathfrak {c}$. Thus, in any model of MA+¬CH we have $\omega_{1}<\mathfrak {p}=\mathfrak {c}$. However in a model in which $\omega_{1}<\mathfrak {p}=\mathfrak {c}$ we do not necessarily have MA and in fact it is slightly easier to force just $\omega_{1}<\mathfrak {p}=\mathfrak {c}$ than to force MA+¬CH. Thus, we shall first construct a model of $\omega_{1}<\mathfrak {p}=\mathfrak {c}$, which—by Chapter 13 ∣ Related Result 79—proves the consistency of MA(σ-centred)+¬CH with ZFC, and then we shall sketch the construction of a generic model in which we have MA+¬CH. Finally, we shall consider the case when a single Cohen real c is added to a model V⊨ZFC in which MA+¬CH holds. Even though full MA fails in V[c] (see Related Result 104), we shall see that $\mathfrak {p}=\mathfrak {c}$ still holds in V[c]—a result which will be used in Chapter 27.