What strengthenings of measurability does the Mostowski collapse of the ultrapowers possess? Ok, I already posted this question, but a couple of notational errors and assumptions were made in the previous version. Hopefully, this new equivalent version will be better understood. Let $U$ be a nonprinciple $\lambda$-complete ultrafilter over $\lambda$. Let $\pi_U(f)$ for a function $f$ with domain $\lambda$ be defined as follows: $$\pi_U(f)=\{\pi_U(g):\{\alpha<\lambda:g(\alpha)\in f(\alpha)\}\in U\}$$ Let $M$ be $\{\pi_U(f):\mathrm{Dom}(f)=\lambda\}$. Finally, let $\lambda_0=\lambda$ and: $$\lambda_{n+1}=|\{\pi_U(f):\{\alpha<\lambda:g(\alpha)\in \lambda_n\}\in U\}|$$ **Then, which of the following are always true:** - If $\lambda$ is $\theta$-strong, then $V_\theta\subset M$. - If $\lambda$ is $\theta$-supercompact, then $M^\theta\subset M$. - If $\lambda$ is $n$-superstrong, then $V_{\lambda_n}\subset M$. - If $\lambda$ is $n$-huge, then $M^{\lambda_n}\subset M$.