Notre Dame Journal of Formal Logic

Classical and Intuitionistic Models of Arithmetic

Kai F. Wehmeier


Given a classical theory T, a Kripke structure $\mbox{\bf K} = (K, \leq, (A_{\alpha})_{\alpha \in K})$ is called T-normal (or locally T) if for each $ \alpha \in K $, $ A_{\alpha} $ is a classical model of T. It has been known for some time now, thanks to van Dalen, Mulder, Krabbe, and Visser, that Kripke models of HA over finite frames $ (K, \leq) $ are locally $\mbox{\bf PA}$. They also proved that models of $\mbox{\bf HA}$ over the frame $ (\omega, \leq) $ contain infinitely many Peano nodes. We will show that such models are in fact $\mbox{\bf PA}$-normal, that is, they consist entirely of Peano nodes. These results are then applied to a somewhat larger class of frames. We close with some general considerations on properties of non-Peano nodes in arbitrary models of $\mbox{\bf HA}$.

Article information

Notre Dame J. Formal Logic Volume 37, Number 3 (1996), 452-461.

First available in Project Euclid: 14 December 2002

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03F50: Metamathematics of constructive systems
Secondary: 03C62: Models of arithmetic and set theory [See also 03Hxx] 03F55: Intuitionistic mathematics


Wehmeier, Kai F. Classical and Intuitionistic Models of Arithmetic. Notre Dame J. Formal Logic 37 (1996), no. 3, 452--461. doi:10.1305/ndjfl/1039886521.

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