# Can a Cohen extension have an $\omega_1$-sequence of successively more generic Cohen reals?

A forcing extension $V[g]$ by the Cohen forcing ${\rm Add}(\omega,1)$ has continuum many Cohen generic reals. Indeed, as Joel Hamkins explains in this Math Overflow answer, a Cohen extension has a perfect set of Cohen generic reals such that any two finite subsets of them are mutually generic. Since adding a Cohen real is isomorphic to adding $\omega$-many Cohen reals, a Cohen forcing extension also has countable sequences $\langle r_n\mid n\lt\omega\rangle$ of Cohen reals such that for every $n\lt\omega$, $r_n$ is Cohen generic over $V[\langle r_i\mid i\lt n\rangle]$. Can we push this to obtain a sequence $\langle r_\xi\mid\xi\lt\omega_1\rangle$ such that for every $\alpha\lt\omega_1$, $r_\alpha$ is Cohen generic over $V[\langle r_\xi\mid\xi\lt\alpha\rangle]$? This question arose out of my work with Richard Matthews on ${\rm ZFC}^-$-models and we were stumped on it for some time. My intuition was that the answer should be yes; after all, we are not asking that the sequence is generic for an $\omega_1$-product of Cohen reals. It turns out that Andreas Blass already solved the question years ago and my intuition was all wrong!

Theorem: (Blass ) A Cohen forcing extension cannot have a sequence of Cohen reals $\langle r_\xi\mid\xi\lt\omega_1\rangle$ such that for every $\alpha\lt\omega_1$, $r_\alpha$ is Cohen generic over $V[\langle r_\xi\mid\xi\lt\alpha\rangle]$.

The proof sketch I give here is my modification of Blass's proof which will allow us to generalize the argument to the forcing ${\rm Add}(\kappa,1)$, adding a Cohen subset to $\kappa$, for an inaccessible $\kappa$.

Proof:
Let $\mathbb B$ be the Boolean completion of ${\rm Add}(\omega,1)$. In particular, $\mathbb B$ has a dense subset of size $\omega$. Now suppose that a forcing extension by $\mathbb B$ (equivalentely ${\rm Add}(\omega,1))$ has a sequence $\langle r_\xi\mid \xi\lt\omega_1\rangle$ of Cohen reals such that for every $\alpha\lt\omega_1$, $r_\alpha$ is Cohen generic over $V[\langle r_\xi\mid\xi\lt\alpha\rangle]$. The model $V[\langle r_\xi\mid\xi\lt\omega_1\rangle]$ is a generic extension of $V$ by a complete subalgebra $\mathbb D$ of $\mathbb B$ by the Intermediate Model Theorem of Solovay . Let's argue that $\mathbb D$ also has a dense subset of size $\omega$. Given a condition $p\in {\rm Add}(\omega,1)$, let $q_p$ be the infima of $b$ in $\mathbb D$ such that $p\leq b$. Each $q_p$ is in $\mathbb D$ by completeness and the conditions $q_p$ are dense in $\mathbb D$. Let $\dot R$ be a $\mathbb D$-name such that it is forced by $1$ that (1) $\dot R$ is an $\omega_1$-sequence of successively more generic Cohen reals and (2) the extension by $\mathbb D$ is equal to the extension $V[\dot R]$. Then it is easy to see that the Boolean values $[[n\in \dot R(\xi)]]$ for $n\lt\omega$ and $\xi\lt\omega_1$ must generate $\mathbb D$. Next, observe that since $\mathbb D$ has a countable dense subset, there must be some $\alpha\lt\omega_1$ such that $\mathbb D$ is generated by the Boolean values $[[n\in \dot R(\xi)]]$ for $n\lt\omega$ and $\xi\lt\alpha$. But this means that if $V[G]$ is $\mathbb D$-generic, then $G$ can be recovered from the sequence $\langle R(\xi)\mid\xi\lt\alpha\rangle\, (R=\dot R_G)$, which contradicts that $R(\alpha)$ is $V[\langle R(\xi)\mid\xi\lt\alpha\rangle]$-generic. $\square$

The proof easily generalizes to give the following result for an inaccessible cardinal $\kappa$.

Theorem: A forcing extension by ${\rm Add}(\kappa,1)$ cannot have a sequence, $\langle A_\xi\mid\xi\lt\kappa^+\rangle$ of subsets of $\kappa$ such that for every $\alpha\lt\kappa^+$, $A_\alpha$ is ${\rm Add}(\kappa,1)$-generic over $V[\langle A_\xi\mid\xi\lt\alpha\rangle]$.

I don't know whether the result holds for every regular cardinal $\kappa$ because the current argument relies on $\kappa^{\lt\kappa}=\kappa$.

Question: Does the above result hold true for any regular cardinal $\kappa$?

Question: Can Blass's theorem be proved using partial orders without reference to Boolean algebras?

## References

1. A. Blass, “The model of set theory generated by countably many generic reals,” J. Symbolic Logic, vol. 46, no. 4, pp. 732–752, 1981.
2. S. Grigorieff, “Intermediate submodels and generic extensions in set theory,” Ann. Math. (2), vol. 101, pp. 447–490, 1975.