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Mathematical fuzzy logic and natural numbers. (English) Zbl 1139.03016
Recently, Grzegorczyk has posed the following problem: When weakening Robinson’s arithmetic $$Q$$ by replacing the binary operations of addition and multiplication by ternary operations that define functions possibly not total, is such a theory still essentially undecidable such as Robinson’s classical arithmetic? The paper gives a positive answer to this problem (the introduced ternary relations do not necessarily define total crisp functions), exploiting mathematical fuzzy logic and formulating this weakened arithmetic as fuzzy arithmetic.

##### MSC:
 03B52 Fuzzy logic; logic of vagueness 03F30 First-order arithmetic and fragments