Standard vs. Nonstandard Distinction: A Watershed in the Foundations of Mathematics

Part of the Synthese Library book series (SYLI, volume 251)


In this paper, I will discuss a conceptual distinction, or a contrast, between two opposing ideas, that has played an extremely important role in the foundations of mathematics. The distinction was first formulated explicitly, though not quite generally, by Leon Henkin in 1950.1 He called it a distinction between the standard and the nonstandard interpretation of higher-order logic. I will follow his terminology, even though it may not be the most fortunate one. One reason for saying that this nomenclature is not entirely happy is that it is not clear which interpretation is the “standard” one in the sense of being a more common one historically. Another reason is that one can easily characterize more than one nonstandard interpretation of higher-order logic, even though Henkin considered only one.2 Furthermore, it turns out that the distinction (rightly understood) is not restricted to higher-order logics.’ Last but not least, it is far from clear how Henkin’s notion of standard interpretation or standard model is related to logicians’ idea of the standard model of such first-order theories as elementary arithmetic, in which usage “standard model” means simply “intended model”.


Arbitrary Function Standard Interpretation Propositional Function Intended Model Nonstandard Interpretation 
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© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  1. 1.Boston UniversityUSA

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