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Transfer principles in nonstandard intuitionistic arithmetic

Summary: Transfer principles in nonstandard
intuitionistic arithmetic
Jeremy Avigad and Jeffrey Helzner
March 6, 2001
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 trans-
fer 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, care-
ful attention to the complexity of our definitions allows us to show that a
certain weak fragment of intuitionistic nonstandard arithmetic is conser-
vative over primitive recursive arithmetic.
1 Introduction
Classical models and theories of nonstandard arithmetic and analysis have long


Source: Avigad, Jeremy - Departments of Mathematical Sciences & Philosophy, Carnegie Mellon University


Collections: Multidisciplinary Databases and Resources; Mathematics