# Minimal Realizability of Intuitionistic Arithmetic and Elementary Analysis

Zlatan Damnjanovic
The Journal of Symbolic Logic
Vol. 60, No. 4 (Dec., 1995), pp. 1208-1241
DOI: 10.2307/2275884
Stable URL: http://www.jstor.org/stable/2275884
Page Count: 34

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## Abstract

A new method of "minimal" realizability is proposed and applied to show that the definable functions of Heyting arithmetic (HA)--functions f such that HA $\vdash \forall x\exists!yA(x, y)\Rightarrow$ for all m, A(m, f(m)) is true, where A(x, y) may be an arbitrary formula of L(HA) with only x, y free--are precisely the provably recursive functions of the classical Peano arithmetic (PA), i.e., the $< \varepsilon_0$-recursive functions. It is proved that, for prenex sentences provable in HA, Skolem functions may always be chosen to be $< \varepsilon_0$-recursive. The method is extended to intuitionistic finite-type arithmetic, HAω0, and elementary analysis. Generalized forms of Kreisel's characterization of the provably recursive functions of PA and of the no-counterexample-interpretation for PA are consequently derived.

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