What strengthenings of measurability does the Mostowski collapse of the ultrapowers possess?
Here's whatOk, I meanalready posted this question, but a couple of notational errors and assumptions were made in the previous version. Let $j:V\rightarrow M$Hopefully, this new equivalent version will be an elementary embedding with critical point $\kappa$ where $P(M)$ holds (given some third order predicate)better understood.
Let $MO$$U$ be the Mostowski collapse of $Ult_U(V)$ for somea nonprinciple $\kappa$$\lambda$-complete ultrafilter $U$. When does $P(MO)$ also hold?
A perfect example is measurability itself. $j$ has critical pointover $\kappa$$\lambda$. Letting Let $P(M)$ be "there is an elementary embedding$\pi_U(f)$ for a function $j_0:V\rightarrow M$$f$ with critical pointdomain $\kappa$", it can$\lambda$ be seen thatdefined as follows: $$\pi_U(f)=\{\pi_U(g):\{\alpha<\lambda:g(\alpha)\in f(\alpha)\}\in U\}$$
Let $M$ holds iffbe $MO$ holds$\{\pi_U(f):\mathrm{Dom}(f)=\lambda\}$. Finally, let $\lambda_0=\lambda$ and: $$\lambda_{n+1}=|\{\pi_U(f):\{\alpha<\lambda:g(\alpha)\in \lambda_n\}\in U\}|$$
But does it hold forThen, which of the following strengthenings of measurability?are always true:
- LetIf $\kappa$ be$\lambda$ is $\theta$-strong. Is, then $V_\theta\subset MO$?$V_\theta\subset M$.
- LetIf $\kappa$ be$\lambda$ is $\theta$-supercompact. Is, then $MO^\theta\subset MO$?$M^\theta\subset M$.
- LetIf $\kappa$ be$\lambda$ is $n$-superstrong, and $j_0:V\rightarrow MO$ be the composition of the mostowski collapse function (between the ultrapower and $MO$)then $\pi$ to the canonical ultrapower embedding$V_{\lambda_n}\subset M$. Is $V_{j_0^n(\kappa)}\subset MO$?
- LetIf $\kappa$ be$\lambda$ is $n$-huge, and $j_0:V\rightarrow MO$ be the composition of the mostowski collapse function (between the ultrapower and $MO$)then $\pi$ to the canonical ultrapower embedding$M^{\lambda_n}\subset M$. Is $MO^{j_0^n(\kappa)}\subset MO$?
