Authors: Marković, Zoran 
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: On the structure of kripke models of heyting arithmetic
Journal: Mathematical Logic Quarterly
Volume: 39
Issue: 1
First page: 531
Last page: 538
Issue Date: 1-Jan-1993
ISSN: 0942-5616
DOI: 10.1002/malq.19930390154
Abstract: 
Since in Heyting Arithmetic (HA) all atomic formulas are decidable, a Kripke model for HA may be regarded classically as a collection of classical structures for the language of arithmetic, partially ordered by the submodel relation. The obvious question is then: are these classical structures models of Peano Arithmetic (PA)? And dually: if a collection of models of PA, partially ordered by the submodel relation, is regarded as a Kripke model, is it a model of HA? Some partial answers to these questions were obtained in [6], [3], [1] and [2]. Here we present some results in the same direction, announced in [7]. In particular, it is proved that the classical structures at the nodes of a Kripke model of HA must be models of IΔ1 (PA‐ with induction for provably Δ1 formulas) and that the relation between these classical structures must be that of a Δ1‐elementary submodel. MSC: 03F30, 03F55.
Keywords: Heyting arithmetic | Kripke model | Peano arithmetic

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