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Suppose $0^{\#}$ exists. Let $M(L)$ be the generic multiverse over $L$ as viewed in $V$, i.e. $M(L)$ consists of all $L[g]$ where $g \in V$ is $\mathbb{P}$-generic over $L$ for some forcing $\mathbb{P} \in L$.

In a previous post we established that $M(L)$ is not upward closed, i.e. there are $M,N \in M(L)$ such that for all $K \in M(L) \colon M \cup N \not \subseteq K$.

Let us consider the following weakening:

Definition 1. $M(L)$'s pointwise theory is reconcilable iff for all $M,N \in M(L)$ there are $M',N' \in M(L)$ such that $M \subseteq M', N \subseteq N'$ and $\mathrm{Th}((M'; \in)) = \mathrm{Th}((N'; \in))$.

Question 2. Is $M(L)$'s pointwise theory reconcilable?

Again, there is an easy partial answer:

Proposition 3. Let $M_{< \omega_{1}}(L)$ be the generic universe over $L$ restricted to posets $\mathbb{P}$ such that $V \models \mathrm{card}(\mathbb{P}) < \omega_1$. $M_{< \omega_{1}}(L)$'s pointwise theory is reconcilable.

Proof. Let $\mathbb P, \mathbb Q \in L$ be such that $$ V \models \mathrm{card}(\mathbb{P}),\mathrm{card}(\mathbb{Q}) < \omega_1. $$ Let $g \in V$ be $\mathbb{P}$-generic over $L$ and let $h \in V$ be $\mathbb{Q}$-generic over $L$. Since $\omega_1^V$ is a limit of $L$-inaccessibles, we may fix some $L$-inaccessible $\kappa < \omega_1^L$ such that $\mathbb{P}, \mathbb{Q} \in \mathcal{J}_\kappa$. Since $\mathcal{J}_{\kappa}$ is countable in $V$, there are $g',h' \in V$ which are $\mathrm{Coll}(\omega, \kappa)$-generic over $L$ such that $L[g] \subseteq L[g']$ and $L[h] \subseteq L[h']$. Now recall that $\mathrm{Coll}(\omega, \kappa)$ is weakly homogeneous and thus $\mathrm{Th}((L[g']; \in)) = \mathrm{Th}((L[h']; \in))$. Q.E.D.

Suppose $0^{\#}$ exists. Let $M(L)$ be the generic multiverse over $L$ as viewed in $V$, i.e. $M(L)$ consists of all $L[g]$ where $g \in V$ is $\mathbb{P}$-generic over $L$ for some forcing $\mathbb{P} \in L$.

In a previous post we established that $M(L)$ is not upward directed, i.e. there are $M,N \in M(L)$ such that for all $K \in M(L) \colon M \cup N \not \subseteq K$.

Let us consider the following weakening:

Definition 1. $M(L)$'s pointwise theory is reconcilable iff for all $M,N \in M(L)$ there are $M',N' \in M(L)$ such that $M \subseteq M', N \subseteq N'$ and $\mathrm{Th}((M'; \in)) = \mathrm{Th}((N'; \in))$.

Question 2. Is $M(L)$'s pointwise theory reconcilable?

Again, there is an easy partial answer:

Proposition 3. Let $M_{< \omega_{1}}(L)$ be the generic universe over $L$ restricted to posets $\mathbb{P}$ such that $V \models \mathrm{card}(\mathbb{P}) < \omega_1$. $M_{< \omega_{1}}(L)$'s pointwise theory is reconcilable.

Proof. Let $\mathbb P, \mathbb Q \in L$ be such that $$ V \models \mathrm{card}(\mathbb{P}),\mathrm{card}(\mathbb{Q}) < \omega_1. $$ Let $g \in V$ be $\mathbb{P}$-generic over $L$ and let $h \in V$ be $\mathbb{Q}$-generic over $L$. Since $\omega_1^V$ is a limit of $L$-inaccessibles, we may fix some $L$-inaccessible $\kappa < \omega_1^L$ such that $\mathbb{P}, \mathbb{Q} \in \mathcal{J}_\kappa$. Since $\mathcal{J}_{\kappa}$ is countable in $V$, there are $g',h' \in V$ which are $\mathrm{Coll}(\omega, \kappa)$-generic over $L$ such that $L[g] \subseteq L[g']$ and $L[h] \subseteq L[h']$. Now recall that $\mathrm{Coll}(\omega, \kappa)$ is weakly homogeneous and thus $\mathrm{Th}((L[g']; \in)) = \mathrm{Th}((L[h']; \in))$. Q.E.D.

Suppose $0^{\#}$ exists. Let $M(L)$ be the generic multiverse over $L$ as viewed in $V$, i.e. $M(L)$ consists of all $L[g]$ where $g \in V$ is $\mathbb{P}$-generic over $L$ for some forcing $\mathbb{P} \in L$.

In a previous post we established that $M(L)$ is not upward closed, i.e. there are $M,N \in M(L)$ such that for all $K \in M(L) \colon M \cup N \not \subseteq K$.

Let us consider the following weakening:

Definition 1. $M(L)$'s pointwise theory is upward directed iff for all $M,N \in M(L)$ there are $M',N' \in M(L)$ such that $M \subseteq M', N \subseteq N'$ and $\mathrm{Th}((M'; \in)) = \mathrm{Th}((N'; \in))$.

Question 2. Is $M(L)$'s pointwise theory upward directed?

Again, there is an easy partial answer:

Proposition 3. Let $M_{< \omega_{1}}(L)$ be the generic universe over $L$ restricted to posets $\mathbb{P}$ such that $V \models \mathrm{card}(\mathbb{P}) < \omega_1$. $M_{< \omega_{1}}(L)$'s pointwise theory is upward directed.

Proof. Let $\mathbb P, \mathbb Q \in L$ be such that $$ V \models \mathrm{card}(\mathbb{P}),\mathrm{card}(\mathbb{Q}) < \omega_1. $$ Let $g \in V$ be $\mathbb{P}$-generic over $L$ and let $h \in V$ be $\mathbb{Q}$-generic over $L$. Since $\omega_1^V$ is a limit of $L$-inaccessibles, we may fix some $L$-inaccessible $\kappa < \omega_1^L$ such that $\mathbb{P}, \mathbb{Q} \in \mathcal{J}_\kappa$. Since $\mathcal{J}_{\kappa}$ is countable in $V$, there are $g',h' \in V$ which are $\mathrm{Coll}(\omega, \kappa)$-generic over $L$ such that $L[g] \subseteq L[g']$ and $L[h] \subseteq L[h']$. Now recall that $\mathrm{Coll}(\omega, \kappa)$ is weakly homogeneous and thus $\mathrm{Th}((L[g']; \in)) = \mathrm{Th}((L[h']; \in))$. Q.E.D.

Suppose $0^{\#}$ exists. Let $M(L)$ be the generic multiverse over $L$ as viewed in $V$, i.e. $M(L)$ consists of all $L[g]$ where $g \in V$ is $\mathbb{P}$-generic over $L$ for some forcing $\mathbb{P} \in L$.

In a previous post we established that $M(L)$ is not upward closed, i.e. there are $M,N \in M(L)$ such that for all $K \in M(L) \colon M \cup N \not \subseteq K$.

Let us consider the following weakening:

Definition 1. $M(L)$'s pointwise theory is reconcilable iff for all $M,N \in M(L)$ there are $M',N' \in M(L)$ such that $M \subseteq M', N \subseteq N'$ and $\mathrm{Th}((M'; \in)) = \mathrm{Th}((N'; \in))$.

Question 2. Is $M(L)$'s pointwise theory reconcilable?

Again, there is an easy partial answer:

Proposition 3. Let $M_{< \omega_{1}}(L)$ be the generic universe over $L$ restricted to posets $\mathbb{P}$ such that $V \models \mathrm{card}(\mathbb{P}) < \omega_1$. $M_{< \omega_{1}}(L)$'s pointwise theory is reconcilable.

Proof. Let $\mathbb P, \mathbb Q \in L$ be such that $$ V \models \mathrm{card}(\mathbb{P}),\mathrm{card}(\mathbb{Q}) < \omega_1. $$ Let $g \in V$ be $\mathbb{P}$-generic over $L$ and let $h \in V$ be $\mathbb{Q}$-generic over $L$. Since $\omega_1^V$ is a limit of $L$-inaccessibles, we may fix some $L$-inaccessible $\kappa < \omega_1^L$ such that $\mathbb{P}, \mathbb{Q} \in \mathcal{J}_\kappa$. Since $\mathcal{J}_{\kappa}$ is countable in $V$, there are $g',h' \in V$ which are $\mathrm{Coll}(\omega, \kappa)$-generic over $L$ such that $L[g] \subseteq L[g']$ and $L[h] \subseteq L[h']$. Now recall that $\mathrm{Coll}(\omega, \kappa)$ is weakly homogeneous and thus $\mathrm{Th}((L[g']; \in)) = \mathrm{Th}((L[h']; \in))$. Q.E.D.

Suppose $0^{\#}$ exists. Let $M(L)$ be the generic multiverse over $L$ as viewed in $V$, i.e. $M(L)$ consists of all $L[g]$ where $g \in V$ is $\mathbb{P}$-generic over $L$ for some forcing $\mathbb{P} \in L$.

In a previous post we established that $M(L)$ is not upward closed, i.e. there are $M,N \in M(L)$ such that for all $K \in M(L) \colon M \cup N \not \subseteq K$.

Let us consider the following weakening:

Definition 1. $M(L)$'s pointwise theory is upward directed iff for all $M,N \in M(L)$ there are $M',N' \in M(L)$ such that $M \subseteq M', N \subseteq N'$ and $\mathrm{Th}((M'; \in)) = \mathrm{Th}((N'; \in))$.

Question 2. Is $M(L)$'s pointwise theory upward directed?

Again, there is an easy partial answer:

Proposition 3. Let $M_{< \omega_{1}}(L)$ be the generic universe over $L$ restricted to posets $\mathbb{P}$ such that $V \models \mathrm{card}(\mathbb{P}) < \omega_1$. $M_{< \omega_{1}}(L)$'s pointwise theory is upward directed.

Proof. Let $\mathbb P, \mathbb Q \in L$ be such that $$ V \models \mathrm{card}(\mathbb{P}),\mathrm{card}(\mathbb{Q}) < \omega_1. $$ Let $g \in V$ be $\mathbb{P}$-generic over $L$ and let $h \in V$ be $\mathbb{Q}$-generic over $L$. Since $\omega_1^V$ is a limit of $L$-inaccessibles, we may fix some $L$-inaccessible $\kappa < \omega_1^L$ such that $\mathbb{P}, \mathbb{Q} \in \mathcal{J}_\kappa$. Since $\mathcal{J}_{\kappa^{+L}}$ is countable in $V$, there are $g',h' \in V$ which are $\mathrm{Coll}(\omega, \kappa)$-generic over $L$ such that $L[g] \subseteq L[g']$ and $L[h] \subseteq L[h']$. Since $\mathrm{Coll}(\omega, \kappa)$ is weakly homogeneous, it follows that $\mathrm{Th}((L[g']; \in)) = \mathrm{Th}((L[h']; \in))$. Q.E.D.

Suppose $0^{\#}$ exists. Let $M(L)$ be the generic multiverse over $L$ as viewed in $V$, i.e. $M(L)$ consists of all $L[g]$ where $g \in V$ is $\mathbb{P}$-generic over $L$ for some forcing $\mathbb{P} \in L$.

In a previous post we established that $M(L)$ is not upward closed, i.e. there are $M,N \in M(L)$ such that for all $K \in M(L) \colon M \cup N \not \subseteq K$.

Let us consider the following weakening:

Definition 1. $M(L)$'s pointwise theory is upward directed iff for all $M,N \in M(L)$ there are $M',N' \in M(L)$ such that $M \subseteq M', N \subseteq N'$ and $\mathrm{Th}((M'; \in)) = \mathrm{Th}((N'; \in))$.

Question 2. Is $M(L)$'s pointwise theory upward directed?

Again, there is an easy partial answer:

Proposition 3. Let $M_{< \omega_{1}}(L)$ be the generic universe over $L$ restricted to posets $\mathbb{P}$ such that $V \models \mathrm{card}(\mathbb{P}) < \omega_1$. $M_{< \omega_{1}}(L)$'s pointwise theory is upward directed.

Proof. Let $\mathbb P, \mathbb Q \in L$ be such that $$ V \models \mathrm{card}(\mathbb{P}),\mathrm{card}(\mathbb{Q}) < \omega_1. $$ Let $g \in V$ be $\mathbb{P}$-generic over $L$ and let $h \in V$ be $\mathbb{Q}$-generic over $L$. Since $\omega_1^V$ is a limit of $L$-inaccessibles, we may fix some $L$-inaccessible $\kappa < \omega_1^L$ such that $\mathbb{P}, \mathbb{Q} \in \mathcal{J}_\kappa$. Since $\mathcal{J}_{\kappa}$ is countable in $V$, there are $g',h' \in V$ which are $\mathrm{Coll}(\omega, \kappa)$-generic over $L$ such that $L[g] \subseteq L[g']$ and $L[h] \subseteq L[h']$. Now recall that $\mathrm{Coll}(\omega, \kappa)$ is weakly homogeneous and thus $\mathrm{Th}((L[g']; \in)) = \mathrm{Th}((L[h']; \in))$. Q.E.D.