This paper presents a type theory in which it is possible to directly manipulate n-dimensional cubes (points, lines, squares, cubes, etc.) based on an interpretation of dependent type theory in a cubical set model. This enables new ways to reason about identity types, for instance, function extensionality is directly provable in the system. Further, Voevodsky’s univalence axiom is provable in t...

INTRODUCTION Intuitionistic mathematics can be structured into two levels. The first level arises directly out of Brouwer's criticism of certain methods and notions of classical mathematics. the law of excluded middle was rejected and instead the meaning of mathematical statements was to be based on the notion of 'proof'. level of intuitionism was a theory of meaning quite different from the cl...

Constructive mathematics is mathematics without the use of the principle of the excluded middle. There exists a wide array of models of constructive logic. One particular interpretation of constructive mathematics is the realizability interpretation. It is utilized as a metamathematical tool in order to derive admissible rules of deduction for systems of constructive logic or to demonstrate the...

We present a generalisation of the type-theoretic interpretation of constructive set theory into Martin-Löf type theory. The original interpretation treated logic in Martin-Löf type theory via the propositions-as-types interpretation. The generalisation involves replacing Martin-Löf type theory with a new type theory in which logic is treated as primitive. The primitive treatment of logic in ty...

The main objective of this paper is to show that a certain formulae-asclasses interpretation based on generalized set recursive functions provides a selfvalidating semantics for Constructive Zermelo-Fraenkel Set theory, CZF. It is argued that this interpretation plays a similar role for CZF as the constructible hierarchy for classical set theory, in that it can be employed to show that CZF is e...