Second‐Order Logic and Mathematics

Stewart Shapiro

in Foundations without Foundationalism

Published in print March 2000 | ISBN: 9780198250296
Published online November 2003 | e-ISBN: 9780191598388 | DOI:
 Second‐Order Logic and Mathematics

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There are a number of central notions that have adequate characterizations in second‐order languages, with no special non‐logical terminology, but which cannot be defined in a first‐order languagedue to the failure of compactness for such languages. The list includes closure, well‐ordering, well‐foundedness, and various cardinalities, such as finitude and denumerability. If these notions are in fact well understood and communicated, as seems plausible, then first‐order languages are inadequate to the task of codifying mathematics. Considerations, some due to Georg Kreisel, are brought to show that first‐order axiomatizations must fail to account for linguistic, semantic, and epistemic features of mathematical practice. The focus is on arithmetic, analysis, and set theory.

Keywords: arithmetic; axiomatization; cardinality; compactness; denumerability; finitude; Kreisel; real analysis; set theory; well‐foundedness; well‐ordering

Chapter.  19098 words. 

Subjects: Philosophy of Mathematics and Logic

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