Chapter

Arbitrary Functions

Mathieu Marion

in Wittgenstein, Finitism, and the Foundations of Mathematics

Published in print August 2008 | ISBN: 9780199550470
Published online September 2011 | e-ISBN: 9780191701559 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199550470.003.0003
Arbitrary Functions

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  • History of Western Philosophy
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In contemporary mathematical logic, the debate over the notion of arbitrary function is reflected in the problem of the interpretation of second-order quantifiers. The distinction between first- and second-order logic hangs on the range of the quantifiers: in first-order logic, quantifiers range uniquely on elements of the structure under study, while in second-order logic, quantifiers can range over the latter's subsets, sets of subsets, and so forth. The range of second-order quantifiers admits of various interpretations from more encompassing or standard ones to more restrictive or non-standard ones. When Leon Henkin made this distinction explicitly in his paper on the ‘Completeness in the Theory of Types’, he considered only one such non-standard interpretation, with his general models, where the higher-order variables are subjected to closure conditions with respect to Boolean and projective operations.

Keywords: arbitrary functions; mathematics; logic; second-order quantifiers; Boolean operations; numerical equivalence; projective operations; extension

Chapter.  15111 words. 

Subjects: History of Western Philosophy ; Philosophy of Mathematics and Logic

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