When a forcing notion adds a Cohen forcing over $V$

Question

Let $\mathbb{P}$ be a forcing notion.

Where I can find what conditions must satisfy $\mathbb{P}$ to add a Cohen real over $ V$?

Someone can give me any reference.

Thank you

Answer 1

This is a great question. I do not know of a full answer that does not simply say that the Cohen algebra completely embeds in the Boolean completion of $\mathbb P$.

There are some nice positive results, though. One that comes up frequently in practice is that any finite support iteration of nontrivial posets adds a Cohen real. This is a serious source of difficulties in the theory of cardinal invariants of the continuum. See for instance

MR1234283 (94h:03102). Goldstern, Martin. Tools for your forcing construction. In Set theory of the reals (Ramat Gan, 1991), 305–360, Israel Math. Conf. Proc., 6, Bar-Ilan Univ., Ramat Gan, 1993.

A reasonably general result can be found in

MR1303493 (96g:03090). Shelah, Saharon. How special are Cohen and random forcings, i.e. Boolean algebras of the family of subsets of reals modulo meagre or null. Israel J. Math. 88 (1994), no. 1-3, 159–174.

There, Shelah proves that any nonatomic Suslin ccc forcing that adds an unbounded real must add a Cohen. Recall that $\mathbb P=(P,\le_P)$ is Suslin if and only if $P$, $\le_P$ and $\bot_P$ are $\mathbf\Sigma^1_1$ sets.

As an example, this gives that the product of any two nonatomic Maharam algebras adds a Cohen. This includes the product of any two nonatomic measure algebras and, as a very particular case, the product of random forcing with itself. The result about Maharam algebras in turn implies that the product of any two nonatomic Suslin ccc posets adds a Cohen real. See

MR2299506 (2008c:03054). Farah, Ilijas; Veličković, Boban. Maharam algebras and Cohen reals. Proc. Amer. Math. Soc. 135 (2007), no. 7, 2283–2290.

This is not an exhaustive list of known positive results, though. But I do not know of a general approach that encompasses all known cases.