Let $\kappa$ be an uncountable regular cardinal, and suppose that $\langle \mathbb{P}_\alpha,\dot{\mathbb{Q}}_\alpha:\alpha<\delta\rangle$ is a $\kappa$-support iteration of $<\kappa$-strategically closed notions of forcing, i.e.,

a) $\Vdash_{\mathbb{P}_\alpha}``\dot{\mathbb{Q}}_\alpha\text{ is $<\kappa$-strategically complete''}$, and

b) inverse limits are taken at every limit stage of cofinality $\leq\kappa$, and direct limits elsewhere.

It is well-known (even folklore) that the limit forcing $\mathbb{P}_\delta$ is also $<\kappa$-strategically closed (see, e.g., Proposition 7.9 of Cummings chapter in the Handbook of Set Theory).

Is there a place where a proof of this appears in the literature?

I am asking because I need to use that the obvious strategy has some additional properties, and if I can find an appropriate reference then I can avoid taking a rather technical detour in the middle of another proof.

  • $\begingroup$ I don't know a reference offhand, but doesn't the straightforward argument just work easily? You fix names for the strategies on each coordinate, and then apply them coordinatewise. It doesn't seem to be that different from the corresponding proof where you drop the "strategically" part. Perhaps one of the additional properties you refer to might be that the strategy is support-preserving; but was there something else? $\endgroup$ – Joel David Hamkins Sep 30 '15 at 21:34
  • $\begingroup$ Yep, the straightforward proof works fine, and that's why the published proofs are all of the form "just like the $\kappa$-closed case''. I was just hoping that I had someplace to point readers rather than writing up the proof and noticing that a couple of other things happen. $\endgroup$ – Todd Eisworth Sep 30 '15 at 22:11
  • $\begingroup$ Support preserving is one thing that I want, The other main thing is that if we're working in an $H(\chi)$ with a fixed well-ordering hanging around, then the obvious strategy is "canonical" if we use the well-ordering to make choices when we have to. $\endgroup$ – Todd Eisworth Sep 30 '15 at 22:33
  • $\begingroup$ I would think this is easy enough to explain in a few sentences or a short paragraph in a research article. Or, if there isn't a suitable reference, an alternative may be to post a definitive answer here, and then cite this very question. $\endgroup$ – Joel David Hamkins Oct 1 '15 at 0:04
  • $\begingroup$ Sure. I was hoping to just be able to point somewhere if it were already written up. $\endgroup$ – Todd Eisworth Oct 1 '15 at 0:23

To finish this question off:

Proposition A.1.6 of the paper "Sheva-Sheva-Sheva: Large Creatures" by Roslanowski and Shelah gives the required proof and explicitly points out much of what I was looking for. The rest is obvious from what is there.

The paper appears in Israel Journal of Mathematics 159(2007), 109-174.

It is number 777 in Shelah's numbering of papers (hence the name!)

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  • $\begingroup$ To elaborate on the last line, "Sheva" is the Hebrew word for seven. $\endgroup$ – Asaf Karagila Oct 15 '15 at 23:02
  • $\begingroup$ But are we meant to refer to this as the Roslanowski-Shelah theorem of 2007? I don't think so, since of course this result was widely known long before that. $\endgroup$ – Joel David Hamkins Oct 15 '15 at 23:06
  • $\begingroup$ That would indeed be silly: the result is folklore, and they include in a section recalling some standard facts. They prove the result for the same reason I asked for a reference: we need that the canonical strategy has some nice properties, and it is convenient to have a place to send a reader wanting to see some details. $\endgroup$ – Todd Eisworth Oct 16 '15 at 14:21

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