Formalized Interpretability in Primitive Recursive Arithmetic

Interpretations are a natural tool in comparing the strength of two theories. In this paper we give a brief introduction to the topic of interpretability and interpretability logics. We will focus on the, so far, unknown interpretability logic of PRA. One research technique will be treated. This technique can be best described as restricting the realizations in the arithmetical semantics. 1 What are interpretations and why study them? How to interpret “Eli, Eli, lama sabachtani”? Let us consider the concept of interpretation in the previous phrase1. What does it actually mean to interpret something. Or more specifically, what do we mean when we say that T interprets some utterance φ of S? Well, in this case T can first translate φ to its own language, then place it in an adequate context and then somehow make sense of it. The mathematical notion of interpretation is somewhat similar. We say that a theory T interprets another theory S whenever there is some translation such that all translated theorems of S become provable in T . We give a precise definition. Throughout this paper we will stay in the realm of first-order logic. Definition 1 K is a relative interpretation of a theory S into a theory T , we write K : T S, whenever the following holds. K is a pair 〈δ, F 〉. The first component, δ, is a formula in the language of T with a single free variable. This formula is used to specify the domain of our interpretation in a sense that we will see right now. The second component, F , is an “Eli, Eli, lama sabachtani” were Jesus’ last words. Some scholars translate this to “My God, my God, why hast thou forsaken me?”. Others read it as “My God, my God, how thou dost glorify me!”. Proceedings of the Eighth ESSLLI Student Session Balder ten Cate (editor) Chapter 1, Copyright c © 2003, Joost J. Joosten 1 Formalized Interpretability in Primitive Recursive Arithmetic easy (primitive recursive) map that sends formulas ψ in the language of S, to formulas F (ψ) in the language of T . We demand for all ψ that the free variables of ψ and F (ψ) are the same. The map F should commute with the boolean connectives, like F (α ∧ β) = F (α) ∧ F (β). Moreover F should relativize the quantifiers to our domain specifier δ. Thus, for example F (∀x α) = ∀x (δ(x)→ F (α)). We think this notion of interpretation is a natural one and comes close to our every day use of the concept of interpretation. And indeed it is a natural tool in comparing the the proof strength of two theories. A first guess to say what it means that some theory T is at least as strong as some other theory S could be the following. Whenever S sees the truth of a formula ψ, T should also be able to see the truth of ψ. But, S and T might speak different languages. This is where the idea of a translation comes in. Of course the translation should preserve some structure. Also it seems unreasonable that T should have the same domain of discourse as S. Taking these considerations into account it comes quite natural to say that T is at least as strong as S whenever T interprets S in the sense of Definition 1. In the mathematical and metamathematical literature the here defined notion of interpretation turns up time and again. Perhaps the most famous example is in the proof of the consistency of non-euclidean geometry. In this proof (see for example [Gre96]) a model for non-euclidean geometry is built in a uniform way inside a model for euclidean geometry. Of course we somehow “know” that euclidean geometry is consistent. This uniform model construction is really nothing but an interpretation. Tarski, Mostowski and Robinson first studied interpretations as a (meta) mathematical tool in a systematic way in [TMR53]. They also used interpretations to determine the undecidability of certain theories. It is not hard to convince oneself that some consistent theory T is undecidable whenever T interprets some essentially undecidable theory S. We say that S is essentially undecidable if S is undecidable and every consistent extension of T in the same language is also undecidable. 2 Formalized interpretability In the previous section we have introduced the mathematical notion of interpretability. We have given some arguments to plea that it is a natural and interesting notion to consider. In this section we will add one more argument to our list. We will see that theories can in a certain way speak about interpretations. This insight will provide us with a simple yet expressive formalism in which large parts of metamathematical practise are expressible.