Suppose that I want to force to add a "single" new subset of $\omega$ and not much else. For example, consider the Cohen forcing consisting of finite partial functions from $\omega$ to 2. The forcing I am interested in is different (in fact not CCC, but Proper), but still the generic set codes a subset of $\omega$. 


Question 1: Suppose that $G$ is $P$-generic over $M$, and there is at least one $r\in(\mathscr{P}(\omega)\cap M[G]) \setminus M$. **Is there a condition $(\*)$ such that**


**$P$ satisfies $(\*)$ if and only if $\forall t\in M[G]\cap \mathscr{P}(\omega) \exists a, b\in M\cap \mathscr{P}(\omega) [ t = (a \cap r) \cup (b\setminus r)] $ ?**


Question 2 (Iterability): Consider a model $M\_\delta$ resulting from an iteration of forcings $P\_\alpha : \alpha<\delta$. Say that each $P\_\alpha$ adds a real $r\_\alpha$. Then $ \{r\_\alpha : \alpha<\delta\} \subseteq \mathscr{P}(\omega)\cap M\_\delta$. **Is there a model $M\_\delta'\subseteq M\_\delta$ such that the reals of $M\_\delta'$ are just $(\mathscr{P}(\omega)\cap M)\cup \{ (a\cap r\_\alpha)\cup (b\setminus r\_\beta) : a,b\in \mathscr{P}(\omega)\cap M; \ \alpha, \beta<\delta\} $ ?**



Forcing to "minimally" add new reals.
There is no forcing that satisfies $(\*)$. Let $r$ be a new real, $r \in M[G]\cap \mathcal{P}(\omega) \setminus M$. Take $t = \{n < \omega \mid n + 1\in r\}$, and let $a, b\in M$ such that $t = (a\cap r)\cup (b\setminus r)$. 


I will show that it is possible to reconstruct $r$ from $a, b$ and the bit $0\in r$. This implies that $r\in M$.


Let $n < \omega$ and assume that we know whether $n \in r$ or not. We want to check if $n + 1\in r$. Split into four cases: 


1. If $n \in a \cap b$ then $n\in t$ and therefore $n + 1\in r$.
2. If $n \notin a \cup b$, $n\notin t\implies n+1\notin r$.
3. If $n \in a\setminus b$ then $n\in t \iff n\in r$ and therefore $n + 1\in r \iff n\in r$.
4. If $n \in b\setminus a$ then $n\in t \iff n\notin r$ and therefore $n + 1\in r \iff n\notin r$.


So we can reconstruct $r$, bit by bit, in $M$ (up to two possibilities) from $a, b$. 


