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Towards metamathematics of weak arithmetics over fuzzy logic. (English) Zbl 1243.03032
Summary: This paper continues the investigation of a very weak arithmetic \(\mathrm{FQ}^{\sim }\) that results from the well-known Robinson arithmetic Q by not assuming that addition and multiplication are total functions (the axiom system \(\mathrm{Q}^{\sim }\)) and, secondly, by weakening classical logic to the basic mathematical fuzzy logic BL\(\forall \) (or to the monoidal t-norm logic MTL\(\forall \)). This investigation was started in the paper [P. Hájek, Fundam. Inform. 81, No. 1–3, 155–163 (2007; Zbl 1139.03016)] where the first Gödel incompleteness of \(\mathrm{FQ}^{\sim }\) (i.e. essential incompleteness) is proved. Here we first discuss \(\mathrm{Q}^{\sim }\) over the Gödel fuzzy logic G\(\forall \), or alternatively over the intuitionistic predicate logic, showing essential incompleteness and essential undecidability; then we prove essential undecidability of \(\mathrm{FQ}^{\sim }\) (correcting an error in [loc. cit.], show a variant of the second Gödel incompleteness theorem for an extension of \(\mathrm{FQ}^{\sim }\) and present a model of the last theory which is fuzzy (non-crisp), has commutative addition and multiplication and non-associative addition.

MSC:
03B52 Fuzzy logic; logic of vagueness
03C62 Models of arithmetic and set theory
03F30 First-order arithmetic and fragments
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