Return to Revisions

Here is an example from a transitive set model of ZFC + "there is a proper class of measurable cardinals". If there is one, then there is a countable one $N$, so fix such an $N$. Fix a sequence $\left<\kappa_n,U_n\right>_{n<\omega}$ of successor measurables $\kappa_n$ of $N$ cofinal in $\mathrm{OR}^N$, with $\kappa_n<\kappa_{n+1}$, and $U_n\in N$ such that $N\models$"$U_n$ is a $\kappa_n$-complete nonprincipal ultrafilter over $\kappa_n$". Let $M$ be the model given by the length $\omega+1$ iteration of $N$, using the (images of the) $U_n$'s. That is, let $M_0=N$ and $E_0=U_0$ and $M_1=\mathrm{Ult}(M_0,E_0)$ and $i_{01}:M_0\to M_1$ the ultrapower map. Given $M_n$ and $i_{0n}:M_0\to M_n$, let $E_{n+1}=i_{0n}(U_n)$ and $M_{n+1}=\mathrm{Ult}(M_n,E_n)$ and $i_{0,n+1}:M_0\to M_{n+1}$ be $j\circ i_{0n}$ where $j:M_n\to M_{n+1}$ is the ultrapower map. Let $M_\omega$ be the direct limit of the $M_n$. Then $M_\omega$ is a transitive model of ZFC, $M_\omega\subseteq N$, but $M_\omega$ is not definable over $N$. In fact, $(N,M_\omega)$ does not model ZFC$_U$, where $U$ is an added predicate interpreted by $M_\omega$. (Is that what you were asking for in the second part?). For the $\kappa_n$'s are exactly those ordinals $\kappa$ which are measurable in $N$ but not measurable in $M_\omega$.