Journal of Symbolic Logic

Minimal Realizability of Intuitionistic Arithmetic and Elementary Analysis

Zlatan Damnjanovic

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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 $\mathscr{L}$(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, $HA^\omega_0$, 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.

Article information

Source
J. Symbolic Logic, Volume 60, Issue 4 (1995), 1208-1241.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183744873

Mathematical Reviews number (MathSciNet)
MR1367206

Zentralblatt MATH identifier
0854.03054

JSTOR
links.jstor.org

Citation

Damnjanovic, Zlatan. Minimal Realizability of Intuitionistic Arithmetic and Elementary Analysis. J. Symbolic Logic 60 (1995), no. 4, 1208--1241. https://projecteuclid.org/euclid.jsl/1183744873


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