Boolean Threshold Representations of Finite and Rational Łukasiewicz Truth Values

Can many-valued truth be constructed from classical Boolean truth? For the finite Łukasiewicz chain, each truth value k/n is uniquely determined by n Boolean answers to threshold questions. We prove that the resulting algebra of thermometer strings is isomorphic to the chain as an MV-algebra, that the representation is dimensionally minimal and essentially unique up to coordinate permutation, and that it extends compositionally to formulas. We then organise the finite algebras into a directed system indexed by divisibility and prove that its direct limit is isomorphic to the rational MV-algebra Q∩[0,1]. As n grows, the construction approximates the standard MV-algebra [0,1] with uniform error O(1/n).