Does $0^{\#}$ imply the failure of upward directedness in the set generic universe over $L$?

Question

Question 1. Suppose that $0^{\#}$ exists. Are there set generic filters $g,h$ over $L$ ($g,h \in V$) such that for any $f$ ($\in V$) which is generic over $L$ we have that $L[g] \cup L[h] \not \subseteq L[f]$?

In other words: Does $0^{\#}$ imply the failure of the upward directedness of the set generic universe over $L$?

I have the following partial result:

Corollary 2. Suppose $0^{\#}$ exists. Let $\kappa$ be the least innaccessible cardinal of $L$. Then there are $g,h$ which are $\mathbb{C}$-generic over $L$ ($\mathbb{C}$ is Cohen forcing) such that for any $f$ which is $\mathbb{P}$-generic over $L$ for some $\mathbb{P} \in J_{\kappa}$ we have $L[g] \cup L[h] \not \subseteq L[f]$.

That observation was inspired by:

Proposition 3. (Mostowski?) Let $M$ be a countable transitive model of $\mathrm{ZFC}$ and let $\mathbb{C}$ be Cohen forcing. Then there are $\mathbb{C}$-generic filters $g,h$ over $M$ such that for any transitive $\mathrm{ZFC}$ model $N$ with the same ordinal height of $M$ we have $M[g] \cup M[h] \not \subseteq N$.

In particular there is no set generic filter $f$ over $M$ such that $M[g] \cup M[h] \subseteq M[f]$.

Proof. Let $\alpha = M \cap \mathrm{Ord} < \omega_{1}$ and fix a bijection $f \colon \omega \to \alpha$. Define $E \subseteq \omega \times \omega$ via $$ (m,n) \in E \iff f(m) \in f(n). $$ Let $z \in ^{\omega}2$ code $E$ in an absolute fashion. E.g. we may let $$ z(k) = 1 \iff k = 2^{m}\cdot3^{n} \wedge (m,n) \in E. $$ Clearly any transitive model having $z$ as an element has $E$ as an element as well and, since $E$ is a well-order of order type $\alpha$, must also have $\alpha$ as an element.

It now suffices to construct Cohen reals $c,d$ over $M$ such that they combined code the real $z$. We construct $c,d \in ^{\omega}2$ as follows:

Since $M$ is countable, we may fix an enumeration $(D_{k} \mid k < \omega)$ of all dense subsets of $\mathbb{C}$ that are elements of $M$. Let $c_{0} \in D_{0}$ and let $d_{0} = (0^{\mathrm{length}(c_{0})}) ^{\frown} (1) ^{\frown} z(0) ^{\frown} y_{0}$ for some $y_{0}$ such that $d_{0} \in D_{0}$. Given $c_{k}, d_{k}$ for some $k < \omega$ we let $$ c_{k+1} = c_{k} ^{\frown} (0^{\mathrm{length}(d_{k})- \mathrm{length}(c_{k})}) ^{\frown} (1) ^{\frown} x_{k+1} $$ for some $x_{k+1}$ such that $c_{k+1} \in D_{k+1}$ and $$ d_{k+1} = d_{k} ^{\frown} (0^{\mathrm{length}(c_{k+1}) - \mathrm{length}(d_{k})}) ^{\frown} (1) ^{\frown} y_{k+1} $$ for some $y_{k+1}$ such that $d_{k+1} \in D_{k+1}$. We then let $c = \bigcup_{k < \omega} c_{k}$ and $d = \bigcup_{k < \omega} d_{k}$. The initial segments of $c$ and $d$ look as follows $$ \begin{array}{cc|c|c|c|c} c = & c_{0} & 0 \ldots 0 & 1 \ x_{1} & 0 \ldots 0 & \ldots \\ d = & 0 \ldots 0 & 1 \ z(0) \ y_{0} & 0 \ldots 0 & 1 \ z(1) \ y_{1} & \ldots \end{array} $$ and the blocks of $0$'s in $c$ and $d$ now allow us to reconstruct $z$ from $c,d$ via a recursive function. (Q.E.D.)

Proof of Corollary 2. Since $0^{\#}$ exists, $\kappa$ exists and is countable in $V$. Since all dense subsets of $\mathbb{C}$ that $L$ can see are in $J_{\omega_{1}^{L}} \subseteq J_{\kappa}$, there are only countably many such dense sets. Just like before we fix a real $z$ that codes the countable ordinal $\kappa$ and construct $g,h$ which are $\mathbb{C}$-generic over $L$ such that they combined code $z$. Suppose there were some $\mathbb{P} \in J_{\kappa}$ and some $f$ which is $\mathbb{P}$-generic over $L$ with $L[g] \cup L[h] \subseteq L[f]$. $g,h,f$ are, of course, $\mathbb{C}$(respectively $\mathbb{P}$)-generic over $J_{\kappa}$ and we would have $g,h \in J_{\kappa}[f]$. But now $J_{\kappa}[f]$ would have to contain $\kappa$ as an element. Contradiction! (Q.E.D.)

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Side note: The proof above only uses that there is some $\kappa$ which is worldly in $L$ but countable in $V$ -- the assumption that $0^{\#}$ exists is overkill. It's just the first setting that came to mind when I thought about an inner model that has a lot of generic filters in $V$.

Answer 1

Note that we can use $g,h$ to code any real of $V$ in a recursive fashion (recursive in the Cohen reals added by $g$ and $h$). In particular we can use them to code $0^{\#}$ -- a real that cannot be added to $L$ via set forcing. This shows that $0^{\#}$ does indeed imply the failure of upward directedness of the set generic universe over $L$.

On the other hand, if $\mathrm{HOD}^V = L$, any set is generic over $L$ (via the Vopenka algebra) and hence the set generic universe over $L$ is upward directed.

Answer 2

One can omit $0^\sharp$, as well as the need for any large cardinals, if one simply limits the class of forcing extensions. For example, perhaps one wants to consider only amalgamation and non-amalgamation in the generic multiverse of Cohen-real extensions $L[c]$, or in the generic multiverse of proper-forcing extensions $L[G]$ or $\omega_1$-preserving forcing extensions.

Theorem. If $\omega_1^L$ is countable in $V$, then there are $L$-generic Cohen reals $c$ and $d$, whose corresponding extensions $L[c]$ and $L[d]$ are not amalgamated by any $L$-generic Cohen real $e$. Indeed, there are such $c$ and $d$ for which there is no $\omega_1$-preserving extension $L[G]$ extending both $L[c]$ and $L[d]$.

Proof. Assume $\omega_1^L$ is countable in $V$. In particular, in $V$ we can enumerate all the dense subsets of Cohen forcing $\text{Add}(\omega,1)$ in $L$ in an $\omega$-sequence in $V$. From this enumeration, we can easily build $L$-generic Cohen reals. Using the proof of proposition 3 in the question, we can code any real coding $\omega_1^L$, and this real cannot be added in any $\omega_1$-preserving forcing over $L$. $\Box$.

Let me also add a mention of our recent paper:

M. E. Habič, J. D. Hamkins, L. D. Klausner, J. Verner, and K. J. Williams, Set-theoretic blockchains, ArXiv e-prints, pp. 1-23, 2018. (under review)

Abstract. Given a countable model of set theory, we study the structure of its generic multiverse, the collection of its forcing extensions and ground models, ordered by inclusion. Mostowski showed that any finite poset embeds into the generic multiverse while preserving the nonexistence of upper bounds. We obtain several improvements of his result, using what we call the blockchain construction to build generic objects with varying degrees of mutual genericity. The method accommodates certain infinite posets, and we can realize these embeddings via a wide variety of forcing notions, while providing control over lower bounds as well. We also give a generalization to class forcing in the context of second-order set theory, and exhibit some further structure in the generic multiverse, such as the existence of exact pairs.