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I think the following question is due to Prikry:

Question. Is it consistent that any non-trivial c.c.c forcing notion adds a Cohen real or a Random real?

Is the question still open? What partial results are known about this question (with references, please).

Remark. I have the following simple observation which might be well-know.

Theorem. Souslin hypothesis (SH) hols iff any non-trivial c.c.c forcing notion adds a new real.

Proof. One direction is trivial, since a Souslin tree, considered as a forcing notion, is c.c.c and adds no new reals.

For the other direction suppose there is a non-trivial c.c.c forcing notion $P$ which adds no new reals. Let $B=R.O(P)$ be the completion of $P$. B is a c.c.c. complete Boolean algebra which is $(\omega, \omega)$-distributive, hence it is in fact $(\omega, \infty)-$distributive, thus it is a Souslin algebra, which implies the existence of a Souslin tree, and hence SH fails.

Update. Can anyone say the first place where the above question has appeared?

Mohammad Golshani
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  • What about "Prikry reals"? Fix an ultrafilter U on $\omega$, and let $(s,A) \in \mathbb{P}$ when $s$ is finite, $A \in U$, etc., same as Prikry forcing. This is obviously c.c.c., so do you know if it is consistent that Prikry real forcing adds a Cohen real or Random real? – Monroe Eskew Nov 03 '13 at 07:20
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    One can show that Mathias forcing relative to an ultrafilter U adds Cohen reals exactly when U is not selective. – Mohammad Golshani Nov 03 '13 at 07:47
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    The question whether every Suslin ccc forcing notion adding a real must add a Cohen real or add a random real is problem 4.7 in Shelah's "On what I do not understand": http://shelah.logic.at/files/666.pdf – Haim Nov 03 '13 at 11:22
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    Check also the paper "Ccc forcing and splitting reals" by Velickovic, according to which every Suslin ccc forcing adds a Cohen real or is a Maharam algebra. It's also worth mentioning that the question whether every Maharam algebra is a measure algebra was answered negatively by Talagrand. – Haim Nov 03 '13 at 11:26
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    I believe the theorem in your question was proved by Jim Baumgartner. I know that it was proved by me, quite some time ago, but then I was told that Jim had proved it considerably earlier yet. – Andreas Blass Nov 19 '13 at 18:39
  • Thanks a lot for you historical information. I think the theorem is just a simple observation, and I arrived to it, not by thinking on the above problem, but on "Foreman's maximality principle" which says "any non-trivial forcing notion either adds a real or collapses some cardinals". Note that a trivial consequence of this principle is that any non-trivial c.c.c. forcing adds a real. – Mohammad Golshani Nov 20 '13 at 04:03

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