4
$\begingroup$

I am looking for examples in the literature of countable support iterations of ccc (particularly $\sigma$-centered) forcings, possibly with some emphasis on iterations that avoid adding Cohen reals.

Alternatively, I would also be interested in reasons why such iterations are "uninteresting", for reasons other than the usual limitations on cs iteration.

Improve this question
$\endgroup$
4
  • $\begingroup$ Don't $\sigma$-centered po-sets always add a Cohen? Or is that an open question? I vaugely remember something about the existence of a certain type of ultrafilter being required. $\endgroup$
    – Not Mike
    Commented Dec 28, 2017 at 21:41
  • $\begingroup$ @NotMike If $\mathcal{U}$ is a selective ultrafilter, then the restricted Mathias (or Laver) forcing with side conditions in $\mathcal{U}$ is $\sigma$-centered, but does not add Cohen reals. See, e.g.: mathoverflow.net/questions/96930/…. These are the sorts of forcings I'm thinking about. $\endgroup$
    – Iian Smythe
    Commented Dec 29, 2017 at 0:48
  • 4
    $\begingroup$ I wouldn't call such iterations "uninteresting". Miyamoto's thesis may have results along these lines. In particular, I think he showed the countable support iteration of Cohen forcing of weakly compact length forces $\omega_2$ has the tree property. $\endgroup$
    – Jing Zhang
    Commented Dec 30, 2017 at 15:09
  • $\begingroup$ @IianSmythe Thank you for clearing up my confusion. $\endgroup$
    – Not Mike
    Commented Jan 6, 2018 at 19:33

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .