Towards metamathematics of weak arithmetics over fuzzy logic

This paper continues investigation of a very weak arithmetic FQ^∼ that results from the well-known Robinson arithmetic Q by not assuming that addition and multiplication are total functions (the axiom system Q^∼) and, secondly, by weakening the classical logic to the basic mathematical fuzzy logic BL∀ (or to the monoidal t-norm logic MTL∀). This investigation was started in the paper [5] where the first Gödel incompleteness of FQ^∼ (i.e. essential incompleteness) is proved. Here we first discuss Q^∼ over the Gödel fuzzy logic G∀, or alternatively over the intuitionistic predicate logic, showing essential incompleteness and essential undecidability; then we prove essential undecidability of FQ^∼ (correcting an error in [5]), show a variant of the second Gödel incompleteness theorem for an extension of FQ^∼ and present a model of the last theory which is fuzzy (non-crisp), has commutative addition and multiplication and non-associative addition.