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I've seen in a couple of places (here and here (top of p.26)) that if S is effectively interpretable in T and T is decidable, then S is decidable. We know that first-order logic in a signature with identity and at least one relation symbol is undecidable (see here). But let T be the theory of linear orders without end points and let S $= \emptyset$ (both in the signature with $=$ and $<$). Then the identity mapping is an effective interpretation of S into T, T is decidable, but S isn't.

What am I missing?

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I'd say that it's most likely a mistake on part of the authors of those documents, and it's only true if the interpretation is exact (it's not standard terminology as far as I know) in the sense that an $S$-sentence is true iff its interpretation in $T$ is true (and not only in the one direction). The two definitions are, however, equivalent if $S$ is complete, and that's probably the source of the confusion (maybe they implicitly assume completeness? Or maybe not implicitly, I didn't read through the entirety of either of those documents).

Otherwise, it would imply that any undecidable theory is essentially undecidable (since any theory is clearly interpretable in the weak sense by its decidable extension, if it exists...), but that's clearly not the case, as your example shows.

tomasz
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  • I have never heard it put this way. These seem to refer to specific models though. The definition for interpretability on wiki is purely syntactic – UserB1234 Mar 14 '14 at 02:15
  • @DanulG: when talking about interpreting of one model in another, then the theories can be reasonably assumed to be complete (as theories of models). – tomasz Mar 14 '14 at 02:16
  • @DanulG: I'm not sure what you mean. I'd check Marker's Model Theory. – tomasz Mar 14 '14 at 02:24
  • Sorry I deleted my last comment. What you seem to be saying is that the consequences of $T$ are the same as those of $S$. But then they have the same set of consequences and the entire discussion sort of becomes trivial. – UserB1234 Mar 14 '14 at 02:28
  • @DanulG: they are not the same, as the signatures of $T$ is different from that of $S$. For example, (a model of) set theory interprets the group of integers and not the other way around, while the ring of integers and the semiring of natural numbers interpret one another, but their theories are quite distinct. – tomasz Mar 14 '14 at 02:34
  • OK. I see what you mean. Thank you. I missed that the signatures could be different. OP was talking about the case where they had the same signature. – UserB1234 Mar 14 '14 at 02:38
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    @DanulG: Actually, the signatures can be even (formally) the same, as in the case of naturals vs integers, but their meanings can still be different (like addition for naturals in integers is only defined for naturals, not for negative integers, while for integers in naturals it is actually defined on a subset of ${\bf N}^2$ under the most common interpretation). – tomasz Mar 14 '14 at 02:42
  • This looks tricky. You and I seem to be viewing the problem with different viewpoints, in the sense that there are several different ways in which the interpretation is made, specially if the signatures are different. For example suppose you have a model $M$ of set theory . The ideal way to interpret PA in the model would be to interpret the symbols in the natural number of $M$. But it is not the only way and I'm not sure how you would do this. I would like to see a place where some conventions are established if at all possible. – UserB1234 Mar 14 '14 at 02:50
  • @tomasz That's right. If we have the biconditional "S proves $\phi$ iff T proves $\tau(\phi)$", then the claim is true. Similarly, if T is consistent and S is complete. But I'm pretty sure at least one of links I posted wasn't assuming these stronger conditions. Let me investigate and get back to you! –  Mar 14 '14 at 06:59
  • @DanulG: Well, you could interpret it in some nonstandard or uncountable model. About your question regarding conventions, truthfully, I don't really know. I do and I've been taught model theory, but never needed to know more about this specific subject than I've understood intuitively. But I'd wager there should be something in one of the good model theory textbooks such as those shared in this thread: http://math.stackexchange.com/q/161924/30222 . – tomasz Mar 14 '14 at 09:30
  • I don't think there is anything there in the textbooks. Different forms of undecidability can be passed around using techniques that interpret a model of one theory as a model of an other theory. But there are certain requirements for this and I don't think that the iff in your answer encapsulates these requirements. The proof theoretic way is probably better. But all that does is essentially say that the consequences are the same (Your "interpreting" assumes that even if the signatures are different you can think of one signature as obtained by the other through an expansion by definitions) – UserB1234 Mar 14 '14 at 14:47
  • Also even if you were to interpret PA in a non-standard model of set theory, you would still want the standard parts to match up. So there is a canonical way in which the interpretaion should be done (for example you would want at least the atomic diagram of the models preserved under different interpretations and you would want one signature to be the expansion of another in some reasonable way, maybe the addition of new constants, or an expansion by definitions) – UserB1234 Mar 14 '14 at 14:48