Do indiscernibility embeddings exist for an initial segment of an inner model of many measurable cardinals?

2011-10-09T23:36:39Z

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.

Answer by Philip Welch for Do indiscernibility embeddings exist for an initial segment of an inner model of many measurable cardinals?

2012-02-17T00:10:11Z

My reading of this question was different from Andreas', because Norman asked for order preserving maps of the indiscernibles to extend to embeddings $j:L[U]_\theta \rightarrow$ $L[U] _\theta$

i.e. the indiscernibles should be below $\theta$ as the ordinal height of the structures mentioned is $\theta$?

In that case the measurable or Ramsey above $\theta$ only guarantees indiscernibles above $\theta$ and so "missing the target"?

In any case there are generalisations of "double mice" that you surmise that provide a positive answer. Let $M$ be the "least" in a certain canonical well-ordering of all such iterable structure that have a measurable cardinal $\kappa$ which is, in $M$, the limit of measurables cardinals below $\kappa$. (Such structures are called "mice".) By Loewenheim Skolem we may assume that $M$ is countable. `Iterability' here means we may form all iterated ultrapowers of the first structure $M=M_0$ using this top measure repeatedly; call the ultrapower structures $M_\tau$ for all ordinal $\tau$; then all the $M_\tau$ will be wellfounded. In the $\tau$'th model let $\kappa_\tau$ be the top-most measurable cardinal. Then $\tau ,\mu \rightarrow \kappa_\tau<\kappa_\mu$ and the class of all such $\kappa_\tau$ forms a closed unbounded class of indiscernibles for the model $W$ "left behind" by these iterates.

[Fornally $W = \bigcup _ \tau H(\kappa_\tau)^{M_\tau}$.]

Then $W$ is the `least' inner model with a proper class of measurables cardinals, (because as $\tau$ increases the order type of the measurables in the models $M_\tau$ increases; but these measures are left behind in the lower $H(\kappa_\tau)^{M_\tau}$ part of the model that goes into $W$). The $\kappa_\tau$ are indiscernibles for it, just as the Silver indiscernibles are for $L$.

The large cardinal assumption is then that "This iterable $M$ exists" just as a parallel to "$0$-sharp exists" is to $L$.

Do indiscernibility embeddings exist for an initial segment of an inner model of many measurable cardinals? - MathOverflow

Do indiscernibility embeddings exist for an initial segment of an inner model of many measurable cardinals?

Answer by Philip Welch for Do indiscernibility embeddings exist for an initial segment of an inner model of many measurable cardinals?