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We have the notations $L[A]$ for the inner-model constructible relative to some $A$, and the notation $M[G]$ for a generic extention of the model $M$. Do they coincide? That is, if we look at the constructible inner model $L$ and we have some $G$ which is a generic subset of some forcing notion $P\in L$, is the generic extention the same as the model constructed relative to $G$? (I think it is but I want to make sure...)

In particular, do we have a Condensation Lemma for $L[G]$?

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Yes, the two things coincide. Simply note that if $G\subseteq L$, then $L[G]$ is the smallest model of $\sf ZFC$ such that $G$ is an element of it.

Generally speaking, $L[G]$ is the smallest model such that $G\cap L[G]\in L[G]$, but if $G\subseteq L$, then we get this automatically.

You can also think about this as $M[G]$ being a model for a language augmented by a predicate, and we interpret the names from $M$ to define this model. This is the same as adding a predicate to the language and considering the definable subsets in the case of forcing over $L$, simply because every name will appear at some point and we will interpret it "correctly".

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    $\begingroup$ Yes, but very blatantly what we mean by $M[G]$ is what in the constructibility sense is actually $L(M\cup\{G\})$. In particular, $L[G]$ would be $L(G)$. They coincide, because $G\in L[G]$, but the notation was erroneously picked. (Just as $L(\mathbb R)$ is written $L[\mathbb R]$ in several classical papers.) $\endgroup$
    – Andrés E. Caicedo
    Apr 14, 2016 at 13:40
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    $\begingroup$ It gets confusing when forcing over $\mathsf{HOD}$, where we want to distinguish between $\mathsf{HOD}_a$ and $\mathsf{HOD}[a]$, and the forcing extension over $\mathsf{HOD}$ with generic $a$. $\endgroup$
    – Andrés E. Caicedo
    Apr 14, 2016 at 13:41
  • $\begingroup$ And it gets much much worse if one is trying to read Grieforieff's paper about intermediate models. $\endgroup$
    – Asaf Karagila ♦
    Apr 14, 2016 at 15:08

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