Do indiscernibility embeddings exist for an initial segment of an inner model of many measurable cardinals? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T06:24:03Z http://mathoverflow.net/feeds/question/77632 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/77632/do-indiscernibility-embeddings-exist-for-an-initial-segment-of-an-inner-model-of Do indiscernibility embeddings exist for an initial segment of an inner model of many measurable cardinals? Norman Lewis Perlmutter 2011-10-09T23:36:39Z 2012-02-17T00:10:11Z <p><strong>Background</strong></p> <p>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.</p> <p><strong>Hypotheses</strong></p> <p>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 $\theta$.</p> <p><strong>Question statement</strong></p> <p>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 $j:L[\mathcal{U}]_\theta \to L[\mathcal{U}]_\theta \, \, ?$</p> <p><strong>Remarks</strong></p> <p>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. </p> <p>In <i>The Core Model</i>, 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.</p> http://mathoverflow.net/questions/77632/do-indiscernibility-embeddings-exist-for-an-initial-segment-of-an-inner-model-of/88674#88674 Answer by Philip Welch for Do indiscernibility embeddings exist for an initial segment of an inner model of many measurable cardinals? Philip Welch 2012-02-17T00:10:11Z 2012-02-17T00:10:11Z <p>My reading of this question was different from Andreas', because Norman asked for order preserving maps of the indiscernibles to extend to embeddings $j:L[U]_\theta \rightarrow$ $L[U] _\theta$ </p> <p>i.e. the indiscernibles should be below $\theta$ as the ordinal height of the structures mentioned is $\theta$?</p> <p>In that case the measurable or Ramsey above $\theta$ only guarantees indiscernibles above $\theta$ and so "missing the target"? </p> <p>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 $\tau ,\mu \rightarrow \kappa_\tau&lt;\kappa_\mu$ and the class of all such $\kappa_\tau$ forms a closed unbounded class of indiscernibles for the model $W$ "left behind" by these iterates.</p> <p>[Fornally $W = \bigcup _ \tau H(\kappa_\tau)^{M_\tau}$.]</p> <p>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 $L$.</p> <p>The large cardinal assumption is then that "This iterable $M$ exists" just as a parallel to "$0$-sharp exists" is to $L$.</p>