If by a size preserving model we mean any bijection between any two elements of it is an element of it. Then:



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> is it a thoerem of $\sf ZFC$ that for any theory $T$ any two equinumerous size preserving models of $T$ are isomorphic? That is, there is a bijection between the domains of those models that preserves the relational sets in those models.
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Are equinumerous size preserving models of a theory isomorphic?
There is no version of this question I can think of which has an affirmative answer. Let $\alpha,\beta$ be distinct countable ordinals such that $L\_\alpha\equiv L\_\beta\equiv L\_{\omega\_1^L}$ (which exist by downward Lowenheim-Skolem + condensation). Then $L\_\alpha\not\cong L\_\beta$ (since distinct levels of $L$ are non-isomorphic), $L\_\alpha$ and $L\_\beta$ are equinumerous (each is countable), and since $L\_{\omega\_1^L}$ is locally countable (= it sees that each of its infinite elements is in bijection with $\omega$) both $L\_\alpha$ and $L\_\beta$ are locally countable as well and so a fortiori satisfy your size preserving condition. Now take $T=Th(L\_{\omega\_1^L})=Th(L\_\alpha)=Th(L\_\beta)$ (and note that this even gives a counterexample where $T$ is **complete**).


