The Functions Provable by First Order Abstraction

Abstract Function provability in higher-order logic is a versatile and powerful framework for conceptual classification as well as verification and derivation of declarative programs. Here we show that the functions provable in second-order logic with first-order set-abstraction are precisely the elementary functions. This holds regardless of whether the logic is classical, intuitionistic, or minimal. The notion of provability here is not purely logical, as it incorporates a trivial theory of data, with axioms stating that each data object has a detectable main constructor which can be destructed. We show that this is necessary, by proving that without such rudimentary axioms the provable functions are merely the functions broadly-represented in the simply typed lambda calculus, a collection that does not even include integer subtraction.

Authors

• Daniel Leivant

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