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I'm currently reading about models of Set Theory, and I'm working on exercises to better understand the concepts. In Kunen's most recent Set Theory text, he mentions that if we have a transitive model $M$ and if $\cap^M$ is defined, then $\cap$ is absolute. Then, he gives the following exercise:

Describe a two-element non-transitive $M$ that is isomorphic to $\{0,1\}$, such that $\cap^M$ is defined, but $\cap$ is not absolute for $M$, and such that $\subset$ is not absolute for $M$.

I'm having trouble coming up with such a two-element $M$ since I haven't seen many examples of models being used outside the theory. Any help would be greatly appreciated!

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Let $M$ consist of two sets $\{A,B\}$, such that $A\not\subset B$, but also $A\in B$ and $B\notin A$. For example, one can take $A=\{\emptyset\}$ and $B=\{A\}$. It follows that $M$ thinks $A$ has no elements (since it has no elements in $M$, and therefore $M\models A\subset B$, even though this isn't true externally to $M$. Similarly, $M$ thinks $A\cap B=A$, since there are no elements in $M$ that are in both $A$ and $B$, and so this intersection has the same elements in $M$ as $A$ has in $M$. But externally, we can see that $A\cap B\neq A$. So $M$ thinks $A$ is empty and $B$ is singleton $A$, so $M$ is isomorphic to $\{0,1\}$.

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  • $\begingroup$ That is a really nice, simple example that gets the job done. Thank you! $\endgroup$ – josh Oct 10 '13 at 2:42

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