
13A Note On Hájek, Paris And Shepherdson's TheoremLogic Journal of the IGPL 13 (2): 261266. 2005.We prove a settheoretic version of Hájek, Paris and Shepherdson's theorem [HPS00] as follows: The set ω of natural numbers must contain a nonstandard natural number in any natural Tarskian semantics of CŁ0, the set theory with comprehension principle within Lukasiewicz's infinitevalued predicate logic. The key idea of the proof is a generalization of the derivation of Moh ShawKwei's paradox, which is a Russelllike paradox for manyvalued logic

4On the crispness of and arithmetic with a bisimulation in a constructive naive set theoryLogic Journal of the IGPL 22 (3): 482493. 2014.

38Comprehension contradicts to the induction within Łukasiewicz predicate logicArchive for Mathematical Logic 48 (34): 265268. 2009.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 MValgebra

73Distinguishing nonstandard natural numbers in a set theory within Łukasiewicz logicArchive for Mathematical Logic 46 (34): 281287. 2007.In ${\mathbf{H}}$ , a set theory with the comprehension principle within Łukasiewicz infinitevalued predicate logic, we prove that a statement which can be interpreted as “there is an infinite descending sequence of initial segments of ω” is truth value 1 in any model of ${\mathbf{H}}$ , and we prove an analogy of Hájek’s theorem with a very simple procedure

12Forcing indestructibility of MAD familiesAnnals of Pure and Applied Logic 132 (2): 271312. 2005.Let A[ω]ω be a maximal almost disjoint family and assume P is a forcing notion. Say A is Pindestructible if A is still maximal in any Pgeneric extension. We investigate Pindestructibility for several classical forcing notions P. In particular, we provide a combinatorial characterization of Pindestructibility and, assuming a fragment of MA, we construct maximal almost disjoint families which are Pindestructible yet Qdestructible for several pairs of forcing notions . We close with a detaile…Read more

98On Evans's Vague Object from Set Theoretic ViewpointJournal of Philosophical Logic 35 (4): 423434. 2006.Gareth Evans proved that if two objects are indeterminately equal then they are different in reality. He insisted that this contradicts the assumption that there can be vague objects. However we show the consistency between Evans's proof and the existence of vague objects within classical logic. We formalize Evans's proof in a set theory without the axiom of extensionality, and we define a set to be vague if it violates extensionality with respect to some other set. There exist models of set the…Read more