The collection of sets hereditarily definable from the ground model and generic reals

Question

I would like to know whether the following definition of $M$ makes sense.

Let $L$ be an arbitrary (uncountable) set and denote by $\mathbb{C}_L$ the poset that adds a family $C:=\{ c_\ell : \ell \in L\}$ of Cohen reals (i.e. its conditions are finite partial functions from $L\times\omega$ into $\omega$, ordered by $\supseteq$).

In the generic extension $V^{\mathbb{C}_L}$, define $M$ as "the collection of sets hereditarily definable from finite sequences of members of C". In Horowitz and Shelah this seems to be understood as the collection of sets hereditarily definable with parameters in the ground model $V$ and in $C^{<\omega}$. Now I ask:

1. Does this notion of $M$ in Horowitz and Shelah makes sense?

   I tend to think it doesn't, in the same way as the class of definable sets is not definable in ZFC. However, I do not see how to justify my suspicions.

2. If the answer to the above is negative, what is the correct interpretation of "the collection of sets hereditarily definable from finite sequences of members of C"?

   Considering other references, the natural interpretation would be something like HOD[C] (the collection of sets hereditarily definable from the ordinals and $C^{<\omega}$), but I would like to ask somebody more familiar with this topic to confirm this to me.

Any help will be very appreciated.

Answer 1

This is indeed a reasonable worry to have, since the obvious way to define $M$ is to go through a truth definition for the whole universe, which can't be done. And certainly the class of parameter-freely definable sets can't be defined in a parameter-free way (otherwise, we would be able to express "the least undefinable ordinal").

However, it turns out that there is a trick we can use to talk about definability inside the structure in at least some situations. This is discussed in detail in Kunen's book, in Chapter V.$2$; the crucial ingredient is the Reflection Theorem, which lets us shift from looking for "global" definitions to "local" definitions, and these pose no problem.

This trick only works when ordinals are allowed as parameters; fortunately, this is the case here (note that every ordinal is in $V$). Incidentally, it's worth noting that $M$ is exactly $HOD(V\cup C^{<\omega})^{V^{\mathbb{C}_L}}$ since $Ord\subset V$.