Let us consider $M\subseteq V$ a transitive class such that $M^\kappa\subseteq M$ for some cardinal $\kappa$. Now take $\mathbb{P}$ a $\kappa^+$-cc (or $\leq\kappa-$distrivutive) forcing notion in $V$. Then it is the case that, provided that $G\subset \mathbb{P}$ is $V$-generic, then $V[G]$ also thinks that $M[G]^\kappa\subseteq M[G]$. I would be interested in to know a detailed proof of both facts because I have founded them mentioned everywhere but, unfortunattely, without any kind of explanations.


Thank you all in advance.



Closure properties of transitive classes when forcing with $\kappa^+$-cc forcing
First, let me give an incorrect argument:


Working in $V[G]$, suppose $X\_\eta$ ($\eta\in\kappa$) is a $\kappa$-sequence of *sets* in $M[G]$. Then we get a $\kappa$-sequence of *names* $\nu\_\eta$ in $M$ which correspond to those sets: $\nu\_\eta[G]=X\_\eta$. Since $M^\kappa\subseteq M$, we have $(\nu\_\eta)\_{\eta\in\kappa}\in M$. But from this and $G$, we can construct the sequence $(X\_\eta)\_{\eta\in \kappa}$ in $M[G]$. So $M[G]^\kappa\subseteq M[G]$.


Why is this wrong? Well, the sequence $X\_\eta$ exists in $V[G]$, not $V$; so $V$ might not see the sequence of names $\nu\_\eta$. If $V$ doesn't see it, then $M$ doesn't have it, and so we're stuck.




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How do we get around this? I'll answer for the $\kappa^+$-c.c. case; hopefully it will be clear how to handle this for the distributive case.


Suppose the theorem fails. Then there is a name $\mu\in V$ for such a sequence of names $\nu\_\eta\in M$ which give a sequence of sets in $M[G]^\kappa$ but not $M[G]$. Now, the key point is that *each $\nu\_\eta$ is in $M$, and hence in $V$!* So what $\nu\_\eta$ is, is forced by some single condition in $G$.


For each $\eta$, let $A\_\eta$ be a maximal antichain of conditions which decide what $\nu\_\eta$ is. By the chain condition on $\mathbb{P}$, $A\_\eta$ has size $\le\kappa$; by the closure condition on $M$, this means $A\_\eta\in M$, and by the closure condition *again* the sequence $\mathcal{A}=(A\_\eta)\_{\eta\in\kappa}$ is in $M$. But from $\mathcal{A}$ and $G$, we can construct the sequence of $X\_\eta$s in $M[G]$: given $\eta$, look for a condition $p\in G$ which forces a value for $\nu\_\eta$, and then evaluate that name.


