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A note on Hájek, Paris and Shepherdson’s theorem. (English) Zbl 1078.03043
Summary: We prove a set-theoretic version of P. Hájek, J. Paris and J. Shepherdson’s theorem [J. Symb. Log. 65, 339–346 (2000; Zbl 0945.03031)] as follows: The set \(\omega\) of natural numbers must contain a nonstandard natural number in any natural Tarskian semantics of CŁ\(_0 (\omega)\), the set theory with comprehension principle within Łukasiewicz’s infinite-valued predicate logic. The key idea of the proof is a generalization of the derivation of Moh Shaw-Kwei’s paradox [J. Symb. Log. 19, 37–40 (1954; Zbl 0055.00503)], which is a Russell-like paradox for many-valued logic.

MSC:
03E70 Nonclassical and second-order set theories
03B50 Many-valued logic
03B52 Fuzzy logic; logic of vagueness
03F30 First-order arithmetic and fragments
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