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Question. Is there any set forcing notion $\mathbb{P}$ in one of the following categories?

(a) $\mathbb{P}$ preserves cardinals but it is still open whether $\mathbb{P}$ adds reals or not.

(b) $\mathbb{P}$ adds no real but it is still open whether $\mathbb{P}$ preserves cardinals or not.

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    The question is related to Foreman's maximality principle, see http://mathoverflow.net/questions/64280/questions-about-aleph-1-closed-forcing-notions – Mohammad Golshani Mar 02 '14 at 10:11
  • @MohammadGolshani, Yes. What I am looking for is the possible consequences of assuming validity of Foreman-Magidor-Shelah dichotomy conjecture for such forcing notions. –  Mar 02 '14 at 10:17
  • The following question of Kunen might be relevant: Force with perfect $P\subseteq 2^\omega$ such that for every $I\in [\omega]^\omega, \pi_I: I \to 2^I$ have a countable range. Is $\omega_1$ collapsed? – Mohammad Golshani Mar 02 '14 at 10:19

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There are certainly plenty of examples for question (a). The notions of forcing $\mathbb{P}$ arise in the context of iterating proper notions of forcing where each iterand is known not to add reals, but we don't understand what happens in the limit.

Some specific examples:

(1) Forcing related to the Moore-Mrowka problem and CH

The notion of forcing isolated in this paper is proper and adds no new reals while accomplishing some topological task. If we let $\mathbb{P}$ be the limit of a countable support iteration of length $\omega_2$ where each iterand is as in the paper, we know $\mathbb{P}$ preserves cardinals (assuming GCH) but it is open if it adds new reals.

The relevant paper is:

Todd Eisworth, Totally proper forcing and the Moore-Mrowka problem, Fund. Math. 177 (2003), no. 2, 121-136.

[The consistency result we were looking for has been obtained in another way, but we still don't understand what happens with this particular notion of forcing.]

(2) Justin Moore's "measuring with CH" problem

The situation here is as in (1), where we have a proper notion of forcing that accomplishes something without adding reals, but we do not know what happens when things are iterated. See section 7.2 of

Todd Eisworth, Justin Tatch Moore, and David Milovich, Iterated forcing and the Continuum Hypothesis, Appalachian Set Theory: 2006-2012, James Cummings and Ernest Schimmerling eds., Chapter 7 207-244, Cambridge University Press (London Mathematical Society Lecture Note Series v. 406) 2013. ISBN:9781107608504

Todd Eisworth
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  • We get a similar situation for your second question when we look at the problem of "properness for larger cardinals". There, we are trying to iterated forcings that are countably closed, and asking if they preserve cardinals. Roslanowski and Shelah have done a lot of work, but this area of set theory still consists mostly of open problems. – Todd Eisworth Mar 03 '14 at 15:23
  • These notions of forcing aren't going to be of much use for Foreman Maximality as they tend to need instances of GCH to be definable. – Todd Eisworth Mar 03 '14 at 15:32
  • Could you perhaps flesh out a bit the matter with these Roslanowski-Shelah forcings (or provide references)? – tci Mar 03 '14 at 22:57
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    @Tanmay: Andrzej has written a very nice survey on the problem of generalizing properness to higher cardinals (including many references to relevant work by him and Shelah): http://www.unomaha.edu/logic/papers/essay.pdf – Haim Mar 05 '14 at 01:19