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 (P, Q). We close with a detailed investigation of iterated Sacks indestructibility.

Abstract. We review three pairwise similar paradoxes, the modest liar paradox, McGee's paradox and Yablo's paradox, which imply the ω-inconsistency. We show that is caused by the fact that co-inductive definitions of formulae are possible because of the existence of the truth predicate.

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Abstract.We investigate what happens if PA LTr2, a co-inductive language, formalizes itself. We analyze the truth concept in fuzzy logics by formalizing truth degree theory in the framework of truth theories in fuzzy logics. Hájek-Paris-Shepherdson's paradox [HPS00] involves that so called truth degrees do not represent the degrees of truthhood (defined by the truth predicate) correctly in Lukasiewicz infinite-valued predicate logic ∀ L, therefore truth degree theory fails there.

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