A note on Hájek, Paris and Shepherdson's theorem

We prove a set-theoretic version of Hájek, Paris and Shepherdson’s theorem [HPS00] as follows: The set ω of natural numbers must contain a non-standard natural number in any natural Tarskian semantics of Cà L0(ω), the set theory with comprehension principle within à Lukasiewicz’s infinitevalued 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.