Forcing, constructibility, and random functions This question is in some ways an offshoot of my recent question about trying to explain forcing to someone (such as Scott Aaronson, whose questions have prompted my questions) encountering it for the first time.  Actually, I have two questions.
In Cohen's book Set Theory and the Continuum Hypothesis, he begins not with an arbitrary countable transitive model of $\mathsf{ZFC}$, but with the minimal model.  That is, Cohen assumes that there exists a set model for $\mathsf{ZFC}$ where the $\in$ relation is the standard one, and $M = L(\alpha)$ for the smallest $\alpha$ such that $M$ is a model of  $\mathsf{ZFC}$ (here $L(\alpha)$ denotes the constructible sets with rank less than $\alpha$).  In this case, the generic extension $M[G]$ can also be described as $L(\alpha,G)$, where $L(\alpha,G)$ is defined in terms of the definable power set operation $\mathscr{D}$:
$$\eqalign{ L(0,G) &:= \lbrace G \rbrace \cup \mathrm{tr\, cl}(G) \cr
   L(\gamma+1,G) &:= \mathscr{D}\bigl(L(\gamma,G)\bigr)\cr
   L(\gamma,G) &:= \bigcup_{\beta<\gamma} L(\beta,G) \quad \mbox{if $\gamma$ is a limit}\cr}$$
Now in general, for any countable transitive model $M$ of $\mathsf{ZFC}$, it is a theorem that $M[G]$ is the smallest transitive model of $\mathsf{ZFC}$ containing both $M$ and $G$.  This brings me to my first question.

For an arbitrary countable transitive model $M$, can $M[G]$ always be described in terms of the definable power set operation?

Suppose now that we are trying to create a model that violates $\mathsf{V}=\mathsf{L}$. We can take our poset $P$ to be the poset of finite partial functions from $\omega$ to $\lbrace 0,1\rbrace$. The standard thing to do now is to take a generic filter $G$ in $P$. Scott wondered whether we could instead take a random function $f$ from $\omega$ to $\{0,1\}$.  That is, for each natural number $n$, we flip a fair coin and set $f(n)=0$ or $f(n)=1$ accordingly.  Given $f$, we can define $G$ to be the set of all restrictions of $f$ to a finite domain; then $G$ is a filter by construction, but $G$ might not be generic.  Now comes the second question.

Will $G$ be $P$-generic over $M$ with positive probability?

At first I thought the answer would be yes, but when I tried to prove it, I realized that I was bumping up against the distinction between measure and category.  It seems that the answer to this question might depend on $M$.  Perhaps for the minimal model $M$ the answer might be yes, and for some other model the answer might be no?
 A: Tim, here are my answers (low on technical but hopefully high on intuition):

*

*The answer to the first question is YES, with one provision. You need to update the
$L(\alpha,G)$ with $L(\alpha,G\cup M)$. Here is the core idea: the minimal model is the constructible universe truncated at $\alpha$, where $\alpha$ is defined by you above. That means that $M$ is made of all constructibe sets from the empty set.

Now, if you throw in G, what do you do? You attempt to build the constructible sets from G (think of the similar notion of relative recursibility. It is , mutatis mutandis, just the same: constructibility is a closure operator on sets).
Onto your question: if you start from $M_0$, a transitive model which is not the minimal one, and you add G, you must add all the constructible sets from G AND M. As it turns out, that set is precisely $M_0[G]$.


*Scott's idea is quite brilliant, basically it can be summed up as generic=random. The comments above are related to it, but not entirely: they talk about a special type of forcing, the so called random forcing, whereas Scott's (and yours ) idea is broader:


is all forcing nothing but some kind of randomness in disguise?

I think the answer is yes and no, it needs to be made precise: what does it mean to "toss a coin"?
One needs to relativise this basic construct to M (remember the story of Cohen entering M? Let us do it too).
Inside M, we can define formally law-like sequences of zeros and 1s, and therefore stipulate that a sequence is random if there is no law-like description of it in M. In this sense, to be made precise, I believe Scott's intuition is correct:
the function which corresponds to the ultrafilter is always M-random.
ADDENDUM: as per Andreas comment below, I think I overstated my claim. Genericity is definitely stronger than just being random. However, I still think that  the other direction, namely that every generic is M-random, still holds.
