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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.

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  • $\begingroup$ What is $\mathfrak{p}$? I don't think I've seen that cardinal characteristic before. $\endgroup$
    – Noah Schweber
    Commented Feb 10, 2017 at 22:04
  • $\begingroup$ @Noah: It's the cardinal characteristics equals to $\frak t$. :-P $\endgroup$
    – Asaf Karagila ♦
    Commented Feb 10, 2017 at 22:14
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    $\begingroup$ 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. $\endgroup$
    – Todd Eisworth
    Commented Feb 10, 2017 at 22:18
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    $\begingroup$ @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. $\endgroup$
    – Jing Zhang
    Commented Feb 10, 2017 at 22:19
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    $\begingroup$ @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. $\endgroup$
    – Todd Eisworth
    Commented Feb 13, 2017 at 15:38

1 Answer 1

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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.)

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    $\begingroup$ 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. $\endgroup$
    – Asaf Karagila ♦
    Commented Feb 11, 2017 at 10:04
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    $\begingroup$ 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 $\aleph_2$. $\endgroup$
    – Goldstern
    Commented Feb 11, 2017 at 11:06
  • $\begingroup$ I suspected that might be the case. $\endgroup$
    – Asaf Karagila ♦
    Commented Feb 11, 2017 at 14:30

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