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Minimal models of Heyting arithmetic

Published online by Cambridge University Press:  12 March 2014

Ieke Moerdijk  and
Erik Palmgren
Ieke Moerdijk
Affiliation:
Mathematical Institute, University of Utrecht, PO Box 80.010, NL-3508 TA Utrecht, The Netherlands E-mail: moerdijk@math.ruu.nl
Erik Palmgren
Affiliation:
Department of Mathematics, Uppsala University, PO Box 480, S-751 06 Uppsala, Sweden E-mail: palmgren@math.uu.se
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Extract

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].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 4 , December 1997 , pp. 1448 - 1460
Copyright
Copyright © Association for Symbolic Logic 1997

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References

REFERENCES

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