By the language of Lukasiewicz logics we understand the algebra of formulas L = 〈L,→,∼〉. The μ-valued Lukasiwicz matrix 〈Aμ, {1}〉, μ = 2, 3, . . ., א0, (cf. [1]), will be denoted here by Mμ. Lμ = R(Mμ) is the set of tautologies of Mμ and is called the μ-valued Lukasiewicz system. All the unexplained notation in this text will come from Wojcicki’s paper [4]. Let us define two versions of א0-valued Lukasiewicz consequence: (A) Cא0(X) is the least set of formulas including X∪ Lא0 and closed under modus ponens. (B) α ∈ C∗ א0(X) iff for every h : L → hom Aא0 we have hα = 1 whenever h(X) ⊆ {1}.