Greets
This exercise, II.9.10 from Kunen's Set Theory (2011 ed.), has bothered me for a time:
Work in $\mathsf{ZFC^-}$ and assume that there is an infinite set $T$ such that $x=\{x\}$ for all $x\in T$. Let $M$ be the class of all $y\in WF(T)$ such that for some finite $A\subseteq T$, $\hat\pi(y)=y$ for all permutations $\pi$ of $T$ such that for some finite $A\subseteq T$, $\pi\upharpoonright A=id_A.$ Prove that $T\in M$, and that $M$ is a transitive model for $\mathsf{ZF^-}$ plus $(\mathsf{AC})^{WF}$ plus "$T$ cannot be well-ordered".
Where $WF(T)$ is the class constructed as follows:
- $R(0,T)=T$
- $R(\alpha+1)=\mathcal P (R(\alpha,T))$
- $R(\gamma,T)=\bigcup_{\alpha<\gamma}R(\alpha,T).$
Let $WF(T)=\bigcup_{\alpha\in ON}R(\alpha,T).$ $\hat\pi$ is the natural extension of $\pi$ to all of $WF(T)$; so that $\forall x,y\in WF(T)[x\in y\leftrightarrow \hat\pi(x)\in\hat\pi(x)].$
I couldn't prove that $M$ is a model of $\mathsf{ZF^-}$, and not even that $M$ is transitive.
However, I made a different contruction yielding the same results:
I considered a sort of $L$ inside $WF(T)$, call it $L(T)$; the same construction of $L$ but beginning with $T$ instead of $\emptyset$. Then you get a rank for each element of $L(T)$.
Then you prove by induction, on the rank, that for each $x\in L(T)$, there is some finite $A\subseteq T$ which is fixed by all $\hat\pi$ such that $\pi\upharpoonright A=A(\clubsuit).$ And also one proves that $\hat\pi$ is surjective restricted on each level, onto the same level, whenever $\pi$ is a permutation of $T$ $(\star)$.
Now one can easily see that $L(T)$ is transitive; for any $\alpha$, $L(\alpha,T)\subseteq L(\alpha+1,T)$, and that comprehension holds in $L(T)$; using $(\star)$. As the construction is with levels, one can show that for any set $y$ with $y\subseteq L(T)$ there is $z\in L(T)$ such that $y\subseteq z$. Putting toghether these facts one shows that $L(T)$ is a model of $\mathsf{ZF^-}$.
Using $(\clubsuit)$ one proves that $T$ cannot be well-ordered in $L(T)$; and as $(WF)^{L(T)}=WF$, we obtain $(AC)^{WF}$.
Now, is the model $M$ given in the exercise the same as the one I constructed?; recalling some model theory this sounds plausible, but I can't see why.
Can the exercise be done in a more direct way?
Thanks