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The following is a well-know fact:

Theorem The real $r$ is Cohen over $V$ iff if it does not belong to any meager Borel set coded in $V$.

Now suppose that $\kappa$ is an uncountable cardinal and let $(r_i: i<\kappa)$ be a sequence of reals.

Question. Is there a characterization theorem as above for the sequence $(r_i: i<\kappa)$ to be $Add(\omega, \kappa)$-generic over $V$, where $Add(\omega, \kappa)$ is the Cohen forcing for adding $\kappa$-many Cohen reals.

If there is a known characterization, giving a reference is welcome.

Remark. I know there are characterizations of Cohen algebra, for example the one given in the paper Characterizations of Cohen algebras, but I'm interested in a characterization parallel to the one given in the above theorem.

Remark 2. The reason I'm asking the question is the following: In the paper Adding many random reals may add many Cohen reals I showed that forcing with $R(\kappa) \times R(\kappa)$ adds a generic filter for $Add(\omega, \kappa)$ (where $R(\kappa)$ is the usual forcing for adding $\kappa$-many random reals), which generalizes the well-known fact that forcing with $R \times R$ adds a Cohen real (where $R$ is the random forcing). The usual proof of the above known result uses characterization of Cohen reals given in the above theorem, while I presented a direct proof without using any characterization result. If there is a characterization as asked above, there might be a different proof parallel to the known one.

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  • $\begingroup$ Can we just say that amalgam of any countably many $r_i$ is a Cohen real? $\endgroup$
    – Monroe Eskew
    Aug 15, 2017 at 15:24
  • $\begingroup$ @MonroeEskew yes I know this which follows from the c. c. c-ness of the forcing and that adding countably many Cohen's is the same as adding just one. It also seems it is enough to get the stated result above but I'm still interested in seeing an answer without using such tricks. $\endgroup$
    – Mohammad Golshani
    Aug 15, 2017 at 16:49
  • $\begingroup$ If we view $Add(\omega,\kappa)$ as the p.o. of finite partial functions $\kappa \rightarrow 2$, then the most natural guess would be that $r \in 2^\kappa$ is $Add(\omega,\kappa)$-generic over $V$ if and only if $r$ is not in any meager Borel subset of $2^\kappa$ coded in $V$. Do you know that this is wrong? It seems to me like the usual proof should go through, but I feel like I must be missing something (because surely you would have tried this already). $\endgroup$
    – Will Brian
    Sep 5, 2017 at 13:12
  • $\begingroup$ @AndresCaicedo: What about Will Brian's guess as stated in his comment? Is it in fact wrong? If so, then what is the counterexample? $\endgroup$
    – Thomas Benjamin
    Sep 6, 2017 at 12:19
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    $\begingroup$ @Thomas That's not how @ works, see here. Anyway, I do not understand why you address the question to me, and why you would guess the result is wrong. I suggest you study Kunen's paper first, and if you still have specific technical objections to the result, then post a new question. (That Handbook has many interesting chapters, by the way. I particularly like Todorcevic's chapter on trees, and the chapter by Baumgartner on the proper forcing axiom.) $\endgroup$
    – Andrés E. Caicedo
    Sep 6, 2017 at 18:42

1 Answer 1

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Chapter 20 of the Handbook of Set Theoretic Topology ("Random and Cohen reals" by Ken Kunen, pp 887-911) deals with such questions.

Quoting from Truss's review:

"Quite a proportion of the paper is devoted to a study of the properties of Cohen extensions of a countable transitive model by $2^I/\mathcal{I}$, where I is an arbitrary index set and $\mathcal{I}$ is the natural "lifting'' to $\mathcal{P}(I)$ of one of the ideals under consideration."

More specifically, Lemma 3.8 on page 903 gives a general result akin to what Will Brian suggested in the comments above.

3.8 Lemma

Let $\mathcal{I}$ be a reasonable ideal. Let $M$ be a countable transitive model for ZFC with $I\in M$. Let $F\in 2^I$. Then $F$ is $\mathcal{I}$-generic over $M$ if and only if $F$ is not in any $M$-coded Baire set in $\mathcal{I}$.

Recall that the Baire sets are defined as belonging to the $\sigma$-algebra generated by the clopen sets. The paper is written from an abstract point of view so there's a lot of notation, but the intent is to lay out what Cohen and Random forcing have in common, and why they can be lifted to large index sets.

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