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Is there an example of a strongly proper ccc forcing that is not equivalent to Cohen forcing?

Monroe Eskew
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  • Shooting from the hip, coding a subset of $\omega_1$ into a real? – Asaf Karagila Dec 07 '16 at 13:49
  • I think this one fails. The conditions are $(s,F)$, where $s$ is a finite subset of $\omega$ and $F$ is a finite set of almost-disjoint subsets of $\omega$ from a given collection of $\omega_1$ many. Let $M \prec H_\theta$ be countable, and let $p \in M$ be a condition. For any $q = (s,F) \leq p$ such that there is $A \in F \setminus M$, there cannot be a reduction of $q$ to $M$. Because if $r \in M$, then $r$ there is $(t,G) \leq r$ in $M$ such that $t$ has some elements of $A$ above $\max s$, so is incompatible with $q$. – Monroe Eskew Dec 07 '16 at 15:43
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    I think "strongly proper" might stand for different things across the literature. Monroe means "strongly proper" in Mitchell's sense. I.e. for club many countable $M$ and every $p \in M$ there is a $p' \le p$ which is an $(M,P)$-strong master condition; which in turn means that for every $p'' \le p'$ there is a $p''|M \in M \cap P$ such that all extensions of $p''|M$ in $M \cap P$ are compatible with $p''$. – Sean Cox Dec 07 '16 at 18:20
  • @Sean: I've never heard "strongly proper" before, and the first Google result was your slides from a talk not long ago. And now I also saw that you two posted a paper on arXiv today. in any case, I didn't know that this definition existed or had different variants. But I did say that I shot from the hip with my suggestion (taking your slide about forcing with side conditions as something which can sometimes be strongly proper, and I figured that coding subsets is in some sense a Cohen forcing with side conditions, so it might just work). – Asaf Karagila Dec 07 '16 at 18:48
  • @AsafKaragila my comment wasn't in response to your suggestion, I merely thought the question needed some clarification of terminology. – Sean Cox Dec 07 '16 at 19:55
  • If there is a strongly proper ccc forcing which does not add a Cohen real, then, then should answer Prikry's well-known question. This is because such a forcing does not add Random reals. For Prikry's question see Forcing with c.c.c forcing notions, Cohen reals and Random reals. – Mohammad Golshani Dec 08 '16 at 04:51
  • @Mohammad: I think that you're reading the question too literally with the last suggestion. I think the meaning of "Cohen forcing" here is adding any number of reals, not just one. – Asaf Karagila Dec 08 '16 at 08:20
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    @MohammadGolshani all (nontrivial) strongly proper forcings add Cohen reals, since there will be countable models $M$ from $V$ such that $G \cap M$ is generic over $V$ for $P \cap M$ (where $P$ is the strongly proper forcing and $G$ is $(V,P)$-generic). – Sean Cox Dec 08 '16 at 20:15
  • @AsafKaragila right, thanks for clarification – Mohammad Golshani Dec 09 '16 at 05:59
  • @SeanCox That's nice, thanks for noting this. – Mohammad Golshani Dec 09 '16 at 06:00

1 Answers1

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The answer to your question is yes, and it follows from the following paper of Koppelberg and Shelah Subalgebras of Cohen algebras need not be Cohen.

In this paper, it is shown, for each $\kappa \geq \aleph_2,$ there exists a non-Cohen complete subalgebra of $Add(\omega, \kappa)$.

This subalgebra, is c.c.c and strongly proper, as a projection of Cohen forcing, but is not isomorphic to Cohen forcing.

Mohammad Golshani
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    Natural follow-up question: Does there exist a strongly proper ccc forcing of uniform density $\aleph_1$ that is not Cohen? – Monroe Eskew Dec 10 '16 at 19:27
  • @MonroeEskew About the follow-up question: perhaps this paper of Shelah and Zapletal is of some relevance. – tci Dec 12 '16 at 12:00
  • Also, these examples are all where for club many models $M$, $1_{\mathbb P}$ is a strong master condition. I wonder if this is a necessary feature of strongly proper ccc forcing. – Monroe Eskew Dec 12 '16 at 16:25
  • The paper in @tci 's comment, Shelah and Zapletal's Embeddings of Cohen algebras, has updated link https://arxiv.org/abs/math/9502230 – David Roberts Sep 26 '21 at 00:56