The dissertation is in the first place a treatment of mathematical interpretations. Interpretations themselves will be studied, but also shall they be used to study formal theories.
Interpretations, when used in comparing theories, tell us, in a natural way, something about proof-strength of formal theories. Roughly, an interpretation j of a theory T into a theory S is a structure-preserving map, mapping axioms of T to theorems of S. Structure-preserving means that the map should com-mute with proof constructions and with logical connectives. For example, the constraints on the map should exclude the possibility that we simply map all axioms of T to some tautology of S, say 1 = 1. The notion of interpretahil- ity that is studied in the thesis, is the notion of relativized interpretability as studied by Tarski et al. in [TMR53].
In the thesis, only interpretations between first order theories of some mini-mal strength will be considered. As said before, interpretations will be used to study theories, but also shall they be the subject of our study. In the latter case, the emphasis lies on the structural behavior of interpretahility. This behavior is manifested in so-called interpretability logics. The dissertation is organised in three parts.
In the first part we introduce the notion of interpretability, and relate it to other important meta-mathematical notions. This results in the well-known Orey-Hajék characterizations of interpretability. Central notions in these characterizations are consistency statements, definable cuts and II1-conservativity.
Interpretability, being a purely syntactical notion, is formalized, and also the formalizations take place in a completely formalized setting. For every implication in the characterizations, we will carefully spell out the conditions on the meta-theory. The characterizations get an especially elegant form when formulated in terms of category theory.
At the end of the first part we will focus on interpretahility logics. In es-pecially, we shall be interested in a modal characterization of IL(All), the interpretability logic of all reasonable arithmetical theories. We present a new principle R for this logic and prove its arithmetical validity. This correctness is proved in two different ways. We present two modal systems corresponding to these different proof methods. We see that all principles in IL(All), known so far, are provable in both modal systems, again, giving rise to two different soundness proofs.
In the section part of the thesis we fully concentrate on the modal semantics for interpretability logics. A central question is the modal completeness of interpretability logics. We give completeness proofs for the logics IL, ILM, ILM0, ILW and ILW*. The completeness proofs for ILM0 and for ILW* can he seen as first proofs. We also expose some applications of completeness proofs.
We try to develop a uniform format for completeness proofs for interpretability logics. However, there still has to be done more work and research in this direction. A step in the good direction is undoubtedly where we introduce full labels and develop some general theory of these labels.
In the last chapter of the second part we prove ILP0W* to be modally incomplete. In this proof, the principle R plays a central role.
In the third and final part of the thesis, we shall study Primitive Recursive Arithmetic (PRA). We shall especially pay some attention to its relation to IΣ1. We shall give two proofs of Parsons' theorem. One proof is of a model-theoretic nature and gives us some insight in provable closure properties of the provably total recursive functions.
PRA and IΣ1 are known to be equi-consistent over PRA. However, IΣ1 prove the consistency of PRA on a definable cut. We give two proofs of this theorem.
An important problem is the modal characterization of IL(PRA), the interpretability logic of PRA. We calculate a frame condition for Beklemishev's principle. We also show that Beklemishev's principle is at least as strong as Zambella's principle.
We give a characterization of the closed fragment of the interpretahility logic of PRA with a constant for IΣ1. Also, we develop a method for calculating up-perbounds on IL(PRA) (and on interpretahility logics in general) and provide such an upperhound. The calculation proceeds by restricting the possibly substitutions in Solovay's completeness proof.
The encore of the thesis deals with modal semantics for the closed fragment of Japapridze's logic GLP. We give a small variant of Ignatiev's model, a model of 'depth ε0'. The model is submitted to a purely modal analysis without further reference to arithmetical ingredients.
[TMR53] A. Tarski, A. Mostowski, and R. Robinson. Undecidable theories. North-Holland, Amsterdam, 1953.