Abstract
A realizability notion that employs only Kalmar elementary functions is defined, and, relative to it, the soundness of EA-(Π10-IR), a fragment of Heyting Arithmetic (HA) with names and axioms for all elementary functions and induction rule restricted to Π10 formulae, is proved. As a corollary, it is proved that the provably recursive functions of EA-(Π10-IR) are precisely the elementary functions. Elementary realizability is proposed as a model of strict arithmetic constructivism, which allows only those constructive procedures for which the amount of computational resources required can be bounded in advance.
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Damnjanovic, Z. Elementary realizability. Journal of Philosophical Logic 26, 311–339 (1997). https://doi.org/10.1023/A:1017994504149
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DOI: https://doi.org/10.1023/A:1017994504149