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Comprehension contradicts to the induction within Łukasiewicz predicate logic. (English) Zbl 1173.03024
The author gives a simpler and shorter proof of Hájek’s theorem that the mathematical induction on \(\omega\) implies a contradiction in the set theory with the comprehension principle within Łukasiewicz predicate logic Ł\(\forall\) [P. Hájek, Arch. Math. Logic 44, No. 6, 763–782 (2005; Zbl 1096.03064)] by extending a related proof given by the author in [Arch. Math. Logic 46, No. 3–4, 281–287 (2007; Zbl 1110.03049)] so as to be effective in any linearly ordered MV-algebra.

MSC:
03B52 Fuzzy logic; logic of vagueness
03E72 Theory of fuzzy sets, etc.
06D35 MV-algebras
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References:
[1] Cantini A.: The undecidability of Grisĭn’s set theory. Studia Logica 74, 345–368 (2003) · Zbl 1039.03040
[2] Hajek P.: On arithmetic in the Cantor-Łukasiewicz fuzzy set theory. Arch. Math. Log. 44(6), 763–82 (2005) · Zbl 1096.03064
[3] Yatabe, S.: Distinguishing non-standard natural numbers in a set theory within Łukasiewicz logic. Arch. Math. Log. (accepted) · Zbl 1110.03049
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