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?

Set Theory, Lemma 15.30). $\endgroup$ – Gabe Goldberg Aug 26 '20 at 18:486more comments