•  38
    Comprehension contradicts to the induction within Łukasiewicz predicate logic
    Archive for Mathematical Logic 48 (3-4): 265-268. 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 MV-algebra
  •  73
    Distinguishing non-standard natural numbers in a set theory within Łukasiewicz logic
    Archive for Mathematical Logic 46 (3-4): 281-287. 2007.
    In ${\mathbf{H}}$ , 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 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
  •  12
    Forcing indestructibility of MAD families
    with Jörg Brendle
    Annals of Pure and Applied Logic 132 (2): 271-312. 2005.
    Let A[ω]ω be a maximal almost disjoint family and assume P is a forcing notion. Say A is P-indestructible if A is still maximal in any P-generic extension. We investigate P-indestructibility for several classical forcing notions P. In particular, we provide a combinatorial characterization of P-indestructibility and, assuming a fragment of MA, we construct maximal almost disjoint families which are P-indestructible yet Q-destructible for several pairs of forcing notions . We close with a detaile…Read more
  •  98
    On Evans's Vague Object from Set Theoretic Viewpoint
    Journal of Philosophical Logic 35 (4): 423-434. 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
  •  13
    A Note On Hájek, Paris And Shepherdson's Theorem
    Logic Journal of the IGPL 13 (2): 261-266. 2005.
    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 CŁ0, 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