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I'm working through the 2013 edition of Kunen's Set Theory, and I'm having some trouble with Exercise II.9.10, which shows the consistency of $ZF$ without Foundation and the negation of the axiom of choice relative to $ZFC$ with Foundation replaced by the assertion that there exists an infinite set of Quine atoms. Replacing Kunen's notation with what I think is more standard, he defines for any transitive set T $$V_0(T)=T \\ V_{\alpha+1}(T)=\mathcal{P}(V_\alpha) \\ V_\gamma(T) = \bigcup_{\beta<\gamma}V_\beta(T) \text{ for limit $\gamma$}$$

and $V(T)$ is the union of $V_\alpha(T)$ over all ordinals. Then, letting T be any infinite set of Quine atoms, he defines a subclass $M$ of $V(T)$ consisting of all $y$ such that there exists a finite $A\subseteq T$ such that for all permutations $\pi$ of $T$ which fix everything in $A$, if $\hat{\pi}$ is the extension of $\pi$ to an automorphism of $V(T)$, then $\hat{\pi}(y)=y$.

It is apparent to me why $M$ satisfies all the set existence axioms of $ZF$, why all the well-founded sets of $M$ can be well-ordered, and why $T$ is amorphous in $M$ and therefore contradicts the well-ordering theorem, but Kunen also claims that M is transitive. However, it seems to me that any permutation of T should induce a permutation of $V_1(T)$, the full power set of T, and so any automorphism of $V(T)$ should fix $V_1(T)$, so $V_1(T)\in M$, letting $A$ be empty. However, $V_1(T)$ contains many sets which are not in $M$, such as all infinite coinfinite subsets of $T$, so $M$ is not transitive and there is no reason to expect Extensionality to hold.

What am I missing here?

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The presentation does seem overly simplistic.

This might be a typo, or an attempt to avoid complex introduction of the subject matter. But $M$ should be all constructed by induction by taking at each successor step $M_{\alpha+1}$ to be those subsets of $M_\alpha$ which are fixed by all automorphisms which fix some finite set of atoms pointwise.

I also feel that presenting this like this robs the reader from learning about permutation models more clearly. And for this topic I would suggest going for Jech "Set Theory" or even "The Axiom of Choice" which is a book dedicated, essentially, to these constructions.

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  • $\begingroup$ Okay, thank you. Rereading the exercise, Kunen refers to M as a set, even though it contains all the ordinals, so an error actually seems pretty likely. $\endgroup$ – Ben Goodman Sep 12 '15 at 20:30

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