I am reading the paper *Ramsey-like cardinals II* by Victoria Gitman and Philip Welch (Journal of Symbolic Logic, vol. 76, no. 2. pp. 541-560, 2011) and maybe I am missing something.

According to the deffinitions provided there:

An $M_0$-ultrafilter $U_0$ is $0$-good if its ultrapower $M_1$ is well-founded.

$U_0$ is $1$-good if it is weakly amenable and $0$-good.

$U_0$ is $2$-good if $U_1=j_{01}(U_0)$ determines a well-founded ultrapower $M_2$ of $M_1$.

Hence, if $U_0$ is $2$-good, we have two well-founded ultrapowers, $M_1$ and $M_2$ and, in general, if $U_0$ is $n$-good we have $n$ well-founded ultrapowers, $M_1,\ldots, M_n$.

However, $U_0$ is $\omega$-good if it provides a sequence of well-founded ultrapowers $M_1,M_2,\ldots$, but the inductive limit $M_\omega$ is not necessarily well-founded. When this happens, $U_0$ is said to be $\omega+1$-good, but the ultrapower $M_{\omega+1}$ must be ill-founded. $U_0$ is $\omega+2$-good if $M_{\omega+1}$ is well-founded, and so on.

So, if I have understood the definition, for $n<\omega$, when $U_0$ is $n+1$-good the ultrapower $M_{n+1}$ is well-founded, but for $\beta\geq \omega$, when $U_0$ is $\beta+1$-good the ultrapower $M_\beta$ is well-founded, but $M_{\beta+1}$ may be ill-founded.

I insist on this because, for instance, in Theorem 4.1 we have an $\alpha$-good ultrafilter, with $\alpha=\beta+1$, and so, if I am right, the ultrapower $M_\beta$ is well-founded, but we cannot say that $M_\alpha$ is also well-founded. However, the proof requires to make a complex construction within $M_\alpha$, and it seems strange to me that nowhere in the text is said something like "notice that $M_\alpha$ maybe ill-founded". In fact, I am not sure about how to handle this situation, since until now I have worked with well-founded ultrapowers only, but before trying to adapt the proof of Theorem 3.11 as indicated, I have preferred to ask here, just in case $M_\alpha$ is well-founded and I am missing why it is so.