Arbitrary Functions
Mathieu Marion · Oxford University Press eBooks · 2008
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.