A Generalization of Takeuti-Gandy Interpretation

The goal of this paper is to present an explanation of the axiom of function extensionality in dependent type theory. This extends naturally to a model of type theory where a type is interpreted by a Kan semisimplicial set and we also present this generalization. The technique for explaining extensionality for simple type theory goes back to Takeuti [24] and Gandy [7]. The basic idea is already present in the introduction of the second edition of Principia Mathematica [20] and an earlier explanation of extensionality can be found in [21]. In general a function may fail to be extensional, i.e. to send equal elements to equal elements; for instance an operator on propositions may fail to preserve logical equivalence. The idea is to consider a relativized model where we quantify over extensional elements. The paper is organized as follows. We first present the Takeuti-Gandy interpretation of simple type theory. Roughly speaking, this interprets a type as a type with an equivalence relation, which is reminiscent of Bishop’s notion of set in constructive mathematics [5]. One can see the notion of Kan simplicial set as a generalization of the notion of type with equivalence relation. We explain some effectivity problems that occur when using the notion of Kan simplicial set as a generalization of type with an equivalence relation. We present then a first semantics, where a type is interpreted as a truncated Kan semisimplicial set of level 6 1. The correctness of this semantics has been formally verified in the system Coq V8.4. The system we interpret is close to the first published version of Martin-Löf type theory [15]. After giving some applications of this semantics, we present a generalization where a type is interpreted as a Kan semisimplicial set. This gives a formal system together with an effective way of transporting structures and properties along any equivalences.