TITLE: Weakly Compact Cardinal and Iterability
QUESTION [7 upvotes]: In $\textit{Set Theory}$ by Jech 1978 edition, in the proof of Lemma 32.5 which you can hopefully see at the Google book link.
In the course of the proof using the tree property, he produces from any weakly compact cardinal $\kappa$ a non principal $L_\alpha$-ultrafilter $U$ on $\kappa$, which is $L_\alpha$-$\kappa$ complete, and moreover the intersection of $\kappa$ many elements of $U$ (taken in $V$) is nonempty. 
The latter implies that countable intersection (taken in $V$) is nonempty. This fact implies iterability for example by 19.13 of Kanamori. So $L_\alpha$ with this ultrafilter can be iterated of length $\text{Ord}$
However, $0^\sharp$ follows from the existence of a mouse (iterable premouse). But $0^\sharp$ has stronger consistency strength than a weakly compact.
Is this $(L_\alpha, U)$ not a pre-mouse? I have found varying definition of pre-mouse. One definition has the addition condition that $\kappa$ should be the largest cardinal of $L_\alpha$. This seems like it may not hold since at the beginning of the proof, the proof chooses an $\alpha$ such that $L_\alpha \models ZF^-$. 
I would like to have the $\text{Ord}$ length iteration for what I am trying to do; however, I am troubled by whether or not this would give an iterable premouse and hence imply sharps. 
Thanks for any clarification you can provide.

REPLY [8 votes]: I would like to elaborate a little on Joel's answer. To carry out the iterated  ultrapower construction with an ultrafilter that is external to the model, as it is in the case of $L_\alpha$ and $U$, requires making extra assumptions. In the case of an ultrafilter that is an element of your model, the successor stages of the construction simply use the image of the ultrafilter from the previous stage. But we cannot apply $j$ to $U$ because $U$ is not an element of $L_\alpha$. To define the successor stage ultrafilter we need that $U$ is "partially internal" to $M$. This translates into the technical requirement that $U$ is weakly amenable: for every $X\in L_\alpha$ such that $|X|^{L_\alpha}=\kappa$, we have $X\cap U\in L_\alpha$. Weak amenability is equivalent to $L_\alpha$ and its ultrapower having the same subsets of $\kappa$, as Joel mentioned. 
It is easy to see that assuming that every $A\subseteq\kappa$ is contained in a transitive model $M$ of size $\kappa$ for which there is a weakly amenable $M$-ultrafilter on $\kappa$ (measuring sets in $M$) with a well-founded ultrapower is a much stronger large cardinal assumption than a weakly compact cardinal. I call such cardinals 1-iterable. Assuming that such weakly amenable $M$-ultrafilter is additionally countably complete characterizes Ramsey cardinals, which imply $0^\sharp$. Meanwhile, it is possible to have $M$-ultrafilters with varying degrees of iterability. An $\alpha$-iterable cardinal is characterized by the existence of weakly amenable $M$-ultrafilter with $\alpha$-many well-founded iterated ultrapowers and $\alpha$-iterable cardinals form a hierarchy of strength below Ramsey cardinals. We introduced and studied these cardinals  together with Philip Welch.