In this paper, we give a constructive nonstandard model of intuitionistic arithmetic (Heyting arithmetic). We present two axiomatisations of the model: one finitary and one infinitary variant. Using the model these axiomatisations are proven to be conservative over ordinary intuitionistic arithmetic. The definition of the model along with the proofs of its properties may be carried out within a constructive and predicative metatheory (such as Martin-Löf's type theory). This paper gives an illustration of the use of sheaf semantics to obtain effective proof-theoretic results.

The axiomatisations of nonstandard intuitionistic arithmetic (to be called HAI and HAIω respectively) as well as their model are based on the construction in [5] of a sheaf model for arithmetic using a site of filters. In this paper we present a “minimal” version of this model, built instead on a suitable site of provable filter bases. The construction of this site can be viewed as an extension of the well-known construction of the classifying topos for a geometric theory which uses “syntactic sites”. (Such sites can in fact be used to prove semantical completeness of first order logic in a strictly constructive framework, see [6].)

We should mention that for classical nonstandard arithmetics there are several nonconstructive methods of proving conservativity over arithmetic, e.g. the compactness theorem, Mac Dowell–Specker's theorem [3].

We give two proofs of strong normalisation for second order classical natural deduction. The first one is an adaptation of the method of reducibility candidates introduced in [9] for second order intuitionistic natural deduction; the extension to the classical case requires in particular a simplification of the notion of reducibility candidate. The second one is a reduction to the intuitionistic case, using a Kolmogorov translation.