# Countable support iterations of proper forcings with fusion (axiom A), that make $\mathfrak{p}$ large

Assume we start with a ground model satifying GCH. What are some proper forcing notions with fusion (satisfying axiom A but not countably closed), which when iterated $\aleph_2$ times with countable support, preserve cardinals and make $\mathfrak{p}=\aleph_2$? Does such a forcing even exist?

For example, the prototypical such forcing, if in the above we replace $\mathfrak{p}$ with:

$\mathfrak{b}$, we get Laver forcing,

$\mathfrak{d}$, we get Miller forcing,

$\mathfrak{c}$, we get Sacks forcing, etc.

Thanks.

• What is $\mathfrak{p}$? I don't think I've seen that cardinal characteristic before. Feb 10, 2017 at 22:04
• @Noah: It's the cardinal characteristics equals to $\frak t$. :-P Feb 10, 2017 at 22:14
• Are you asking for some sort of definable forcing notion? Ccc posets all satisfy axiom A with the auxiliary orderings just equality, but it sounded to me like you want something nicely describable like Laver, Miller, and Sacks. Feb 10, 2017 at 22:18
• @NoahSchweber smallest cardinality of the collection of infinite sets that any finite subcollection has infinite intersection but there is no pseudointersection (containment mod finite) for all of them. Feb 10, 2017 at 22:19
• @Horse The Baumgartner survey where Axiom A is introduced has ccc as the first example, so it's OK for the auxiliary orderings to all be equality. Feb 13, 2017 at 15:38

## 1 Answer

By Bell's theorem, forcing $\mathfrak p = \mathfrak c$ is the same as forcing Martin's axiom for all $\sigma$-centered forcing notions. So the most natural iteration forcing $\mathfrak p = \mathfrak c$ is an iteration (say: of length $\omega_2$) in which each $\sigma$-centered forcing notion (say: of size $<\aleph_2$) appears as an iterand (cofinally often).

Given your explicit requirement for Axiom A forcing notions, another natural candidate is an iteration in which each axiom A forcing appears as an iterand.

(However, neither of these iterations consists of a single forcing notion repeated $\omega_2$ times.)

• How about iterating the lottery sum of all $\sigma$-centered forcings of size $<\aleph_2$? That would surely be considered iterating "the same forcing", and should probably maybe work. Feb 11, 2017 at 10:04
• Perhaps probably maybe not. The number of such forcings is at least $\aleph_2$, so the direct sum of all these forcings would fail to have the $\aleph_2$-cc. A CS iteration of length $\omega_1$ of this sum will collapse $\aleph_2$. Feb 11, 2017 at 11:06
• I suspected that might be the case. Feb 11, 2017 at 14:30