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Minimal realizability of intuitionistic arithmetic and elementary analysis

Zlatan Damnjanovic

Journal of Symbolic Logic 60 (4):1208-1241 (1995)

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Polynomially Bounded Recursive Realizability.Saeed Salehi - 2005 - Notre Dame Journal of Formal Logic 46 (4):407-417.details A polynomially bounded recursive realizability, in which the recursive functions used in Kleene's realizability are restricted to polynomially bounded functions, is introduced. It is used to show that provably total functions of Ruitenburg's Basic Arithmetic are polynomially bounded (primitive) recursive functions. This sharpens our earlier result where those functions were proved to be primitive recursive. Also a polynomially bounded schema of Church's Thesis is shown to be polynomially bounded realizable. So the schema is consistent with Basic Arithmetic, whereas it is (...) Download Export citation Bookmark 2 citations
Theoremizing Yablo's Paradox.Ahmad Karimi & Saeed Salehi - manuscriptdetails To counter a general belief that all the paradoxes stem from a kind of circularity (or involve some self--reference, or use a diagonal argument) Stephen Yablo designed a paradox in 1993 that seemingly avoided self--reference. We turn Yablo's paradox, the most challenging paradox in the recent years, into a genuine mathematical theorem in Linear Temporal Logic (LTL). Indeed, Yablo's paradox comes in several varieties; and he showed in 2004 that there are other versions that are equally paradoxical. Formalizing these versions (...) Download Export citation Bookmark
Elementary realizability.Zlatan Damnjanovic - 1997 - Journal of Philosophical Logic 26 (3):311-339.details 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 (...) Download Export citation Bookmark 3 citations
Provably total functions of Basic Arithemtic.Saeed Salehi - 2003 - Mathematical Logic Quarterly 49 (3):316.details It is shown that all the provably total functions of Basic Arithmetic BA, a theory introduced by Ruitenburg based on Predicate Basic Calculus, are primitive recursive. Along the proof a new kind of primitive recursive realizability to which BA is sound, is introduced. This realizability is similar to Kleene's recursive realizability, except that recursive functions are restricted to primitive recursives. Download Export citation Bookmark 4 citations

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