Abstract
We introduce the simpler and shorter proof of Hajek’s theorem that the mathematical induction on ω implies a contradiction in the set theory with the comprehension principle within Łukasiewicz predicate logic Ł\({\forall}\) (Hajek Arch Math Logic 44(6):763–782, 2005) by extending the proof in (Yatabe Arch Math Logic, accepted) so as to be effective in any linearly ordered MV-algebra.
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References
Cantini A.: The undecidability of Grisĭn’s set theory. Studia Logica 74, 345–368 (2003)
Hajek P.: On arithmetic in the Cantor-Łukasiewicz fuzzy set theory. Arch. Math. Log. 44(6), 763–82 (2005)
Yatabe, S.: Distinguishing non-standard natural numbers in a set theory within Łukasiewicz logic. Arch. Math. Log. (accepted)
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Yatabe, S. Comprehension contradicts to the induction within Łukasiewicz predicate logic. Arch. Math. Logic 48, 265–268 (2009). https://doi.org/10.1007/s00153-009-0127-5
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DOI: https://doi.org/10.1007/s00153-009-0127-5