**Background**

I am interested in elementary embeddings from a model of set theory into itself. One way of producing such elementary embeddings is when the model is generated by indiscernibles; this idea is very closely related to the existence of sharps. Jech and Kanamori discuss $0^{\#}$ and $0^\dagger$ in detail but don't tell me much about other sharps. More advanced resources are difficult to understand without a lot of background knowledge.

**Hypotheses**

Let $\theta$ be an inaccessible cardinal, and suppose that some set $A$ of measurable cardinals below $\theta$ is a stationary subset of $\theta$. For each $\kappa \in A$, let $\mu_\kappa$ be a normal measure on $\kappa$, and let $\mathcal{U} = \{ \langle \kappa, \mu_\kappa \rangle : \kappa \in A \}$. Let $L[\mathcal{U}]_\theta$ denote those elements of $L[\mathcal{U}]$ of rank less than $\theta$.

**Question statement**

Do there exist large cardinal assumptions which imply the existence of a closed unbounded set of ordinal indiscernibles for $L[\mathcal{U}]_{\theta}$ such that every order-preserving map of these indiscernibles extends to an elementary embedding $j:L[\mathcal{U}]_{\theta} \to L[\mathcal{U}]_{\theta} \, \, ?$

**Remarks**

The large cardinal assumptions may be on $\theta$, the elements of $A$, or some other large cardinal. The values of $\theta$, $A$, and the $\mu_\kappa$ may be chosen in whatever way you like subject to the hypotheses above -- I just want this to work in some example, not in every example.

In *The Core Model*, Dodd mentions double mice, a generalization of $0^\dagger$. Maybe some version of these can be used to answer the question affirmatively, but I know nothing about them.

`$0^\#$`

from one measurable cardinal. And my assumption of a measurable cardinal above $\theta$ is probably overkill; a Ramsey cardinal above $\theta$ should suffice, just as for`$0^\#$`

. $\endgroup$`$0^\#$`

into the new context. Note that you can't expect just any old class of indiscernibles to be a club, or any old Ehrenfeucht-Mostowski theory of indiscernibles to be satisfiable by a club. Even in the theory of`$0^\#$`

, Silver needed to get the right E-M theory by taking a class of indiscernibles whose $\omega$-th element is as small as possible. $\endgroup$