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If two theories A and B are mutually interpretable, in the sense of there existing a translation procedure from A to B and from B to A, does it follow that whatever metatheoretic results (e.g., categoricity) hold for one theory hold for the other?

If it is not true generally, does it hold for some subset of metatheoretic results? For instance, I know that mutual interpretability implies equiconsistency (see, e.g., Colin McLarty's second comment here). Intuitively, it seems like categoricity should carry over if it is right to think of failure of categoricity as resulting from a lack of expressive power (like how first order theories often lack categoricity due to an inability to discriminate between different sized infinite models and thus admitting of Skolemized models). Is this much right, at least?

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    $\begingroup$ In general, the answer is not. The remarks at the end of this answer are particularly relevant. $\endgroup$
    – Andrés E. Caicedo
    Dec 30, 2013 at 7:25
  • $\begingroup$ @AndresCaicedo Thanks for the link, it is very helpful. Do you know if my intuition about categoricity (in particular) carrying over is correct or if it, like many hazy intuitions I have, seems plausible only because of its haziness? If not for mutually interepretable theories then for bi-interpretable theories? $\endgroup$
    – Dennis
    Dec 30, 2013 at 7:45
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    $\begingroup$ I would expect no, but I haven't found an example. $\endgroup$
    – Andrés E. Caicedo
    Dec 30, 2013 at 7:51
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    $\begingroup$ Bi-interpretability is different, though, since essentially we are translating between models of the two theories (of the same size), and the translation would not be possible, if one theory is categorical and the other is not. $\endgroup$
    – Andrés E. Caicedo
    Dec 30, 2013 at 7:57

1 Answer 1

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If $T$ is the theory of a field with $2$ elements and $S$ is the theory of fields of characteristic $2$, then $T$ and $S$ are mutually interpretable, $T$ is categorical and $S$ is not categorical.

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  • $\begingroup$ Just to make sure I understand why S is not categorical, is it because there are uncountably many fields of characteristic 2 and assuming a first-order formulation we have Lowenheim-Skolem available to construct a countable model of S? $\endgroup$
    – Dennis
    Dec 30, 2013 at 18:27
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    $\begingroup$ For every $n > 0$ there is a finite field of size $2^n$... $\endgroup$
    – Carl Mummert
    Dec 30, 2013 at 18:38

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