If there exists a measurable cardinal, we can generate a sequence of iterated ultrapowers $\{Ult_U^\alpha(V)\}_{\alpha\in ON}$. If $0^\sharp$ exists, i.e. if there exists an elementary embedding $j:L\longrightarrow L$, we have a (well-founded) ultrapower $Ult_U(L)$ for a weakly amenable $L$-ultrafilter $U$, but is it iterable?, I mean, can we generate the whole sequence of iterated ultrapowers $\{Ult_U^\alpha(L)\}_{\alpha\in On}$?
I know that this is equivalent to the existence of the sequence $\{Ult_U^\alpha(L)\}_{\alpha\in \omega_1}$, and that a sufficient condition is the $L$-ultrafilter being (externaly) countably complete, which is obviously satisfied for $V$, but maybe not for $L$.
EDITED:
After having studied the proof in Kanamori's book I have three related questions:
- I understand the whole proof except a small step, so I will be grateful to anyone that can solve this (I hope) small doubt to me:
In page 280 it is stated that $e_{\alpha\beta}(\kappa_\alpha)=\kappa_\beta$, and the only indication given is: it follows from (i) and by taking $X=\kappa_\alpha$ in (ii).
Well, (i) implies that $e_{\alpha\beta}$ fixes the ordinals below $\kappa_\alpha$, and (ii) provides $\kappa_\beta=e_{\alpha\beta}(\kappa_\alpha)\cap \kappa_\beta$, so $\kappa_\beta\leq e_{\alpha\beta}(\kappa_\alpha)$, but how can we get that $e_{\alpha\beta}(\kappa_\alpha)\leq \kappa_\beta$?
I see that a sufficient condition is that, if we call $\kappa_\tau=\sup_{\alpha<\tau}\kappa_\alpha$ (what would be $\kappa_\tau$ if the ultrapower $M_\tau$ was well-founded), then $i_{0\alpha}(\kappa_\tau)=\kappa_\tau$ for all $\alpha<\tau$.
- I knew the proof given in Jech's Set Theory (Theorem 18.20) that constructs a set of indiscernibles from an elementary embedding $j:L\longrightarrow L$, and it is very similar to the proof in Kanamori's book. Is this similitude just superficial? Can it be proven that the elementary embeddings constructed in Jech's book are associated to a sequence of iterated ultrapowers? Are they the same embeddings constructed in Kanamori's book?
If this is true, in some sense, then $i_{0\alpha}(\kappa_\tau)=\kappa_\tau$ for all $\alpha<\tau$ should be true, since it is true in Jech's construction.
- The proof in Kanamori's book is long because it does not assume that $0^\sharp$ exists. I mean, maybe there is a more direct proof of the existence of a fully iterable ultrapower relying in the existence of Silver indescernibles, etc. and not only in the existence of a non-trivial embedding.