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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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