# Does $0^{\#}$ imply that $L$'s set generic multiverse has irreconcilable theories?

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.

• The result is false with parameters, since we can add the non-amalgamable Cohen reals. So I think he just means the parameter-free theory. With $0^\sharp$, we have a truth predicate for $L$, so we can refer to the theory in $V$. Jul 22, 2018 at 17:10
• @MihaHabič $\mathrm{Th}(M; \in)$ is the parameter-free first order theory of $(M; \in)$. Jul 22, 2018 at 17:10
• Is it clear that if $0^\sharp$ exists then $V$ has an $L$-generic for some uncountable poset? It seems possible that only countable posets have generics and then your proposition covers all of the cases. For example, what happens in $L[0^\sharp]$? Jul 22, 2018 at 17:20
• Perhaps $V$ can build $L$-generics for $\text{Coll}(\kappa,\kappa^+)$ when $\kappa$ is indiscernible? (even uncountable) After all, $V$ can make a list of $\kappa$ many dense sets that suffice. But I don't quite see how to make it through limits in the diagonalization. Jul 22, 2018 at 17:31
• @Miha $V$ has $L$-generics for ${\rm Col}(\omega,<\omega_1^V)$ if $0^\sharp$ exists. This has been known for a while, at least since Hjorth's thesis. Jul 22, 2018 at 17:45