Say $E$ is a $(\kappa,\lambda)$-extender on some model $\mathcal M$ of set theory, and consider the ultrapower embedding $i:\mathcal M\to\text{Ult}(\mathcal M,E)$. Now recall that a **generator** of $E$ is an ordinal $\xi\in[\kappa,\lambda)$ such that $[a,f]\_E\neq\xi$ for every $a\in[\xi]^{<\omega}$. That is, it can't be "approximated from below". Equivalently, $\xi$ is a generator iff $\text{crit}(\sigma\_\xi)=\xi$, where $\sigma\_\xi:\text{Ult}(\mathcal M,E\upharpoonright\xi)\to\text{Ult}(\mathcal M,E)$ is the embedding $\sigma\_\xi[a,f]\_{E\upharpoonright\xi}:=[a,f]\_E$.


Now, are these generators all cardinals of $\mathcal M$, or equivalently, of $\text{Ult}(\mathcal M,E)$? A generator $\xi$ is certainly a cardinal of $\text{Ult}(\mathcal M,E\upharpoonright\xi)$, being a critical point, but I can't exactly see that this should imply it. If it helps, $\mathcal M$ is to me a premouse in the sense of Mitchell-Steel where $E$ is on the $\mathcal M$-sequence, a fine extender sequence.



Are generators of extenders cardinals?
The answer is not necessarily.


Here is one easy way to make a counterexample. Assume the GCH and
let $j:V\to M$ be a $\kappa^+$-supercompactness embedding with
critical point $\kappa$. So $M^{\kappa^+}\subset M$ and $\kappa^{++}<j(\kappa)$.


Consider the collection of functions $f:\kappa^{<\omega}\to\kappa$ in $V$; there are $\kappa^+$ many such functions. By
$\kappa^+$-supercompactness, the pointwise image of this collection
$\{j(f)\mid f:\kappa^{<\omega}\to\kappa\}$ is in $M$ and has size
$\kappa^+$ there. Thus, inside $M$, there is a club $C$ of ordinals in
$\kappa^{++}$, which is absolute between $M$ and $V$, that are closed under these functions, in the
sense that if $\beta\in C$ and $\vec\alpha\in\beta^{<\omega}$ and $j(f)(\vec\alpha)<\kappa^{++}$, then
$j(f)(\vec\alpha)<\beta$. In
particular, $\beta\neq j(f)(\vec\alpha)$, and so every element of $C$ is a generator in your sense for
the induced extender, but none of them is a cardinal in the target model.


