# c.c.c forcing notions and adding minimal generic reals

Is the following statement consistent:

There is no non-trivial c.c.c forcing notion adding a minimal generic real''?

The question is related to Prikry's question: Is it consistent that any non-trivial c.c.c forcing notion adds a Cohen real or a Random real?

See also A special c.c.c forcing notion and adding minimal generic reals where it is shown that the answer to the above question is no, if we assume $CH$ or $MA+\neg CH$ (this follows from the results of Judah-Shelah in Forcing Minimal Degree of Constructibility) Also it is clear that, a model for Prikry's question is also a model for the above mentioned question.

Remark. By a theorem of Shelah, a modified version of Prikry’s conjecture holds in the context of Souslin c.c.c forcing: i.e. every Souslin c.c.c forcing either adds a Cohen real or is a Maharam algebra. Indeed, Shelah showed that any Souslin c.c.c forcing which is not $ω^ω$-bounding adds a Cohen real. See

S. Shelah, How special are Cohen and random forcing. Israel Journal of Math. 88 (1-3), pp. 159-174, (1994).

• Where do Jensen reals (unique solutions to a $\Pi^1_2$ predicate) come into the picture? Is it consistent that they are not minimal, or consistent that they are not c.c.c., or both? – Asaf Karagila Jun 7 '15 at 6:57
• To prove the forcing is c.c.c, we need our model be somehow $L$-like, for example by looking at section 28 of Jech's set theory book (third millennium), $\Diamond$ is used there. Or in Kanovei paper "A countable definable set of reals containing no definable elements" the use of $L$ (see Lemma 6.4 there), is somehow essential. – Mohammad Golshani Jun 8 '15 at 4:17