Transfer principles in nonstandard intuitionistic arithmetic

Archive for Mathematical Logic 41 (6):581-602 (2002)

Authors
Jeremy Avigad
Carnegie Mellon University
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.
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DOI 10.1007/s001530100109
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References found in this work BETA

Interpreting Classical Theories in Constructive Ones.Jeremy Avigad - 2000 - Journal of Symbolic Logic 65 (4):1785-1812.
Developments in Constructive Nonstandard Analysis.Erik Palmgren - 1998 - Bulletin of Symbolic Logic 4 (3):233-272.
Sheaves and Logic.M. P. Fourman, D. S. Scott & C. J. Mulvey - 1983 - Journal of Symbolic Logic 48 (4):1201-1203.
A Model for Intuitionistic Non-Standard Arithmetic.Ieke Moerdijk - 1995 - Annals of Pure and Applied Logic 73 (1):37-51.
Constructive Sheaf Semantics.Erik Palmgren - 1997 - Mathematical Logic Quarterly 43 (3):321-327.

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Citations of this work BETA

A Functional Interpretation for Nonstandard Arithmetic.Benno van den Berg, Eyvind Briseid & Pavol Safarik - 2012 - Annals of Pure and Applied Logic 163 (12):1962-1994.
Forcing in Proof Theory.Jeremy Avigad - 2004 - Bulletin of Symbolic Logic 10 (3):305-333.
Saturated Models of Universal Theories.Jeremy Avigad - 2002 - Annals of Pure and Applied Logic 118 (3):219-234.
Saturated Models of Intuitionistic Theories.Carsten Butz - 2004 - Annals of Pure and Applied Logic 129 (1-3):245-275.

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