×

zbMATH — the first resource for mathematics

Distinguishing non-standard natural numbers in a set theory within Łukasiewicz logic. (English) Zbl 1110.03049
Summary: In \(\mathbb{H}\), a set theory with the comprehension principle within Łukasiewicz infinite-valued predicate logic, we prove that a statement which can be interpreted as “there is an infinite descending sequence of initial segments of \(\omega\)” has truth value 1 in any model of \(\mathbb{H}\), and we prove an analogy of Hájek’s theorem [ibid. 44, No. 6, 763–782 (2005; Zbl 1096.03064)] with a very simple procedure.

MSC:
03E72 Theory of fuzzy sets, etc.
03B52 Fuzzy logic; logic of vagueness
03B50 Many-valued logic
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Cantini A. (2003). The undecidability of Grišin’s set theory. Stud. Log. 74: 345–368 · Zbl 1039.03040
[2] Hájek P. (2001). Metamathematics of Fuzzy Logic. Kluwer, Dordrecht
[3] Hájek P. (2005). On arithmetic in the Cantor-Łukasiewicz fuzzy set theory. Arch. Math. Log. 44: 763–82 · Zbl 1096.03064
[4] Moh S.-K. (1954). Logical paradoxes for many-valued systems. J. Symbolic Log. 19: 37–40 · Zbl 0055.00503
[5] Restall G. (1993). Arithmetic and truth in Łukasiewicz’s infinitely valued logic. Log. Anal. 36: 25–38 · Zbl 0840.03011
[6] Yatabe S. (2005). A note on Hajek, Paris and Shepherdson’s theorem. Log. J. IGPL 13: 261–266 · Zbl 1078.03043
[7] White R.B. (1979). The consistency of the axiom of comprehension in the infinite-valued predicate logic of Łukasiewicz. J. Philos. Log. 8: 509–534 · Zbl 0418.03037
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.