Transfer principles in nonstandard intuitionistic arithmetic |
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Authors: | J Avigad J Helzner |
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Institution: | (1) Department of Philosophy, Carnegie Mellon University, 135 Baker Hall, Pittsburgh, PA 15213, USA. e-mail: avigad@cmu.edu, US |
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Abstract: | Using a slight generalization, due to Palmgren, of sheaf semantics, we present a term-model construction that assigns a model
to any first-order intuitionistic theory. A modification of this construction then assigns a nonstandard model to any theory
of arithmetic, enabling us to reproduce conservation results of Moerdijk and Palmgren for nonstandard Heyting arithmetic.
Internalizing the construction allows us to strengthen these results with additional transfer rules; we then show that even
trivial transfer axioms or minor strengthenings of these rules destroy conservativity over HA. The analysis also shows that nonstandard HA has neither the disjunction property nor the explicit definability property. Finally, careful attention to the complexity
of our definitions allows us to show that a certain weak fragment of intuitionistic nonstandard arithmetic is conservative
over primitive recursive arithmetic.
Received: 7 January 2000 / Revised version: 26 March 2001 / Published online: 12 July 2002 |
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