The question is whether, when you add a Cohen subset to a cardinal $\kappa$, that cardinal becomes a characteristic of the resulting forcing extension $V[G]$. Or can there be strange instances in which the very same model is realized as a Cohen subset forcing extension over different ground models with different cardinals?

To be precise, can it happen that $M[G]=N[H]$, where $M$ and $N$ are transitive models of ZFC and $G$ is $M$-generic for the forcing to add a Cohen subset to some cardinal $\kappa$, that is, using $\text{Add}(\kappa,1)^M$, and $H$ is similarly $N$-generic to add a Cohen subset to some other cardinal $\delta$, using $\text{Add}(\delta,1)^N$?

For a more concrete version of the question, imagine that we have added a Cohen real $c$ and form the extension $M[c]$; could it be that this model might also be realized as $N[A]$ for some other ground model $N$, where $A$ is an $N$-generic Cohen subset of $\omega_1^N$? Note that $M\neq N$ since it must be that $c\in N$ as the higher forcing does not add reals. For my application, I need to understand the case where the two cardinals are both inaccessible cardinals (if not much more). Also, it is not difficult to identify general situations where this kind of thing is impossible. What I really want to know is if it can ever happen at all.

I conjecture that this situation is impossible, and that indeed, when you add a Cohen subset to a cardinal, you have in particular made that cardinal definable, as "the cardinal for which the universe was just obtained by adding a Cohen subset to it".

The question is really a part of the subject known as set-theoretic geology, but it has recently arisen in another project of mine.

justa Cohen subset, i.e. only $Add(\kappa,1)$, or could you allow forcings which generically adjoin a Cohen subset as a by-product of adjoining some other generic object? $\endgroup$3more comments