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Foundations without FoundationalismA Case for Second-Order Logic$
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Stewart Shapiro

Print publication date: 2000

Print ISBN-13: 9780198250296

Published to Oxford Scholarship Online: November 2003

DOI: 10.1093/0198250290.001.0001

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(p.79) 4 Metatheory
Foundations without Foundationalism

Stewart Shapiro (Contributor Webpage)

Oxford University Press

Although second‐order languages, with standard semantics, are sound for standard semantics, they are not compact, and the Löwenheim–Skolem theorems fail. These are consequences of the fact that the usual second‐order axiomatizations of arithmetic and real analysis are categorical: any two models of either theory are isomorphic to each other. It is a corollary of this and Gödel's incompleteness theorem that second‐order logic with standard semantics is inherently incomplete, in the sense that there is no effective, sound and complete axiomatization for it. It is sometimes claimed that the failure of completeness and compactness are the main shortcomings of second‐order logic. In fact, the failures are tied to the crucial strength of second‐order logic, its ability to give categorical characterizations of rich mathematical structures. It is shown that second‐order languages, with Henkin or first‐order semantics, is not sound, but, restricted to faithful interpretations, it is complete and compact, and the Löwenheim‐Skolem theorems hold.

Keywords:   arithmetic, axiomatization, categoricity, compactness, completeness, Gödel, Henkin, meta‐theory, real analysis, Löwenheim–Skolem, soundness

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