# Interpretability and consistency strength

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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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. – Simon Thomas Mar 30 '13 at 15:00
I would not be surprised if the result is for first order theories that interpret Robinson Arithmetic or something like that. – Colin McLarty Mar 30 '13 at 15:25
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. – Colin McLarty Apr 2 '13 at 11:42
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. – Colin McLarty Apr 2 '13 at 11:45

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. – Andrés Caicedo Mar 30 '13 at 15:51