I’d like to discuss my theorem that the collection of models $M[c]$ obtained by adding an $M$-generic Cohen real $c$ over a fixed countable transitive model of set theory $M$ is upwardly countably closed, in the sense that every increasing countable chain has an upper bound.

I proved this theorem back in 2011, while at the Young Set Theory Workshop in Bonn and continuing at the London summer school on set theory, in a series of conversations with Giorgio Venturi. The argument has recently come up again in various discussions, and so let me give an account of it.

We consider the collection of all forcing extensions of a fixed countable transitive model $M$ of ZFC by the forcing to add a Cohen real, models of the form $M[c]$, and consider the question of whether every countable increasing chain of these models has an upper bound. The answer is yes!  (Actually, Giorgio wants to undertake forcing constructions by forcing over this collection of models to add a generic upward directed system of models; it follows from this theorem that this forcing is countably closed.) This theorem fits into the theme of my earlier post, Upward closure in the toy multiverse of all countable models of set theory, where similar theorems are proved, but not this one exactly.

Theorem. For any countable transitive model $M\models \text{ZFC}$, the collection of all forcing extensions $M[c]$ by adding an $M$-generic Cohen real is upward-countably closed. That is, for any countable tower of such forcing extensions
$$M[c_0]\subset M[c_1]\subset \cdots \subset M[c_n]\subset \cdots ,$$
we may find an $M$-generic Cohen real $d$ such that $M[c_n]\subset M[d]$ for every natural number $n$.

Proof. $$Suppose that we have such a tower of forcing extensions $M[c_0]\subset M[c_1]\subset \cdots$, and so on. Note that if $M[b]\subset M[c]$ for $M$-generic Cohen reals $b$ and $c$, then $M[c]$ is a forcing extension of $M[b]$ by a quotient of the Cohen-real forcing. But since the Cohen forcing itself has a countable dense set, it follows that all such quotients also have a countable dense set, and so $M[c]$ is actually $M[b][b_1]$ for some $M[b]$-generic Cohen real $b_1$. Thus, we may view the tower as having the form:
$$M[b_0]\subset M[b_0\times b_1]\subset \cdots \subset M[b_0\times b_1\times \cdots \times b_n]\subset \cdots ,$$
where now it follows that any finite collection of the reals $b_i$ are mutually $M$-generic.

Of course, we cannot expect in general that the real $⟨b_n\mid n<\omega ⟩$ is $M$-generic for $\text{Add}(\omega ,\omega )$, since this real may be very badly behaved. For example, the sequence of first-bits of the $b_n$’s may code a very naughty real $z$, which cannot be added by forcing over $M$ at all. So in general, we cannot allow that this sequence is added to the limit model $M[d]$. (See further discussion in my post Upward closure in the toy multiverse of all countable models of set theory.)

We shall instead undertake a construction by making finitely many changes to each real $b_n$, resulting in a real $d_n$, in such a way that the resulting combined real $d={\oplus }_n{d}_n$ is $M$-generic for the forcing to add $\omega$-many Cohen reals, which is of course isomorphic to adding just one. To do this, let’s get a little more clear with our notation. We regard each $b_n$ as an element of Cantor space $2^{\omega }$, that is, an infinite binary sequence, and the corresponding filter associated with this real is the collection of finite initial segments of $b_n$, which will be an $M$-generic filter through the partial order of finite binary sequences $2^{<\omega }$, which is one of the standard isomorphic copies of Cohen forcing. We will think of $d$ as a binary function on the plane $d:\omega \times \omega \to 2$, where the $n^{th}$ slice $d_n$ is the corresponding function $\omega \to 2$ obtained by fixing the first coordinate to be $n$.

Now, we enumerate the countably many open dense subsets for the forcing to add a Cohen real $\omega \times \omega \to 2$ as $D_0$, $D_1$, and so on. There are only countably many such dense sets, because $M$ is countable. Now, we construct $d$ in stages. Before stage $n$, we will have completely specified $d_k$ for $k<n$, and we also may be committed to a finite condition $p_{n-1}$ in the forcing to add $\omega$ many Cohen reals. We consider the dense set $D_n$. We may factor $\text{Add}(\omega ,\omega )$ as $\text{Add}(\omega ,n)\times \text{Add}(\omega ,[n,\omega ))$. Since $d_0\times \cdots \times d_{n-1}$ is actually $M$-generic (since these are finite modifications of the corresponding $b_k$’s, which are mutually $M$-generic, it follows that there is some finite extension of our condition $p_{n-1}$ to a condition $p_n\in D_n$, which is compatible with $d_0\times \cdots \times d_{n-1}$. Let $d_n$ be the same as $b_n$, except finitely modified to be compatible with $p_n$. In this way, our final real ${\oplus }_n{d}_n$ will contain all the conditions $p_n$, and therefore be $M$-generic for $\text{Add}(\omega ,\omega )$, yet every $b_n$ will differ only finitely from $d_n$ and hence be an element of $M[d]$. So we have $M[b_0]\cdots [b_n]\subset M[d]$, and we have found our upper bound. QED

Notice that the real $d$ we construct is not only $M$-generic, but also $M[c_n]$-generic for every $n$.

My related post, Upward closure in the toy multiverse of all countable models of set theory, which is based on material in my paper Set-theoretic geology, discusses some similar results.