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I have heard there is some fairly recent result showing that whenever theories $T$ and $T'$ have the same consistency strength, then each can interpret the other. I suppose it refers to first order theories, and I do not know exactly what kind of interpretability it uses or what measure of consistency strength.

Can anyone give me the result, or point me in a good direction?

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    $\begingroup$ This is not true in general since it would imply that any two consistent theories were mutually interpretable. However, the complete theory of the field of real numbers cannot be interpreted in the complete theory of the complex field. $\endgroup$
    – Simon Thomas
    Commented Mar 30, 2013 at 15:00
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    $\begingroup$ I would not be surprised if the result is for first order theories that interpret Robinson Arithmetic or something like that. $\endgroup$
    – Colin McLarty
    Commented Mar 30, 2013 at 15:25
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    $\begingroup$ Thanks to all for comments. Phil Ehrlich pointed me the right way privately. Friedman and Visser have shown the implication from mutual interpretability to eqiconsistency is reversible on a certain subtle condition. We can hastily state the theorem as: If $A$ and $B$ are finitely axiomatized and sequential and consistency of $A$ implies consistency of $B$, then $A$ interprets $B$. The sequentiality addresses Simon Thomas's point. The subtlety is the conditions must be provable in EA, arithmetic with $\Delta_0$-induction plus the axiom that exponentiation is total. $\endgroup$
    – Colin McLarty
    Commented Apr 2, 2013 at 11:42
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    $\begingroup$ And consistency must be \emph{cut free consistency}. Visser has shown how sequentiality needs to be fine tuned in EA. Visser gave a clear example to show that if we need PA to prove "if $A$ is cut free consistent then so is $B$", then the result does not follow: PA proves both that $I\Sigma_1$ is (cut free) consistent and that $I\Sigma_2$ is cut free consistent, yet the first does not interpret the second. $\endgroup$
    – Colin McLarty
    Commented Apr 2, 2013 at 11:45

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John Steel spoke at the EFI series at Harvard concerning his ideas on The triple helix, which has to do in part with the interplay of large cardinal strength and the arithmetic interpretability hierarchy. This is not exactly the claim you mention, but there is a family resemblance.

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  • $\begingroup$ Yes, for "natural" extensions $T$ and $T'$ of $\mathsf{ZF}$, the way that in practice we show that they are equiconsistent provides interpretations in both directions. $\endgroup$
    – Andrés E. Caicedo
    Commented Mar 30, 2013 at 15:51
  • $\begingroup$ I have commented above on a proof theoretic result of Friedmana nd Visser. $\endgroup$
    – Colin McLarty
    Commented Apr 10, 2013 at 11:48

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