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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.

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