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Logic Journal of IGPL 2005 13(2):261-266; doi:10.1093/jigpal/jzi017
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© The Author, 2005. Published by Oxford University Press. All rights reserved.

Original Articles

A note on Hájek, Paris and Shepherdson's theorem

Shunsuke Yatabe

Department of Computer and System Engineering, Kobe University, Rokko-dai, Nada-ku, Kobe 657-8501, Japan. E-mail: yatabe{at}kurt.scitec.kobe-u.ac.jp

We prove a set-theoretic version of Hájek, Paris and Shepherdson's theorem [HPS00] as follows: The set {omega} of natural numbers must contain a non-standard natural number in any natural Tarskian semantics of CL0({omega}), the set theory with comprehension principle within Lukasiewicz's infinite-valued predicate logic. The key idea of the proof is a generalization of the derivation of Moh Shaw-Kwei's paradox, which is a Russell-like paradox for many-valued logic.

Key Words: Lukasiewicz's infinite-valued predicate logic, set theory, the comprehension principle, truth predicate, non-standard natural number

Received 9 September 2004. Revised 11 November 2004.


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