On Interpretability in the Theory of Concatenation

We prove that a variant of Robinson arithmetic Q with non-total operations is interpretable in the theory of concatenation TC introduced by A. Grzegorczyk. Since Q is known to be interpretable in that non-total variant, our result gives a positive answer to the problem whether Q is interpretable in TC. An immediate consequence is essential undecidability of TC. 1 Why weak theories, why concatenation? Several versions of Godel, Church, and Rosser theorems state the incompleteness and undecidability of every sufficiently strong recursively axiomatizable (consistent) theory T . The notion of “sufficiently strong” is usually made precise by stipulating that T extends Robinson arithmetic Q, or more generally, that T interprets Q. Robinson arithmetic Q, see [TMR53], is a theory useful from more than one point of view. It is finitely axiomatized and thus can be used in a straightforward proof of undecidability of first-order predicate logic. It is weak, but some richer arithmetics, like I∆0, are interpretable in it. A natural question reads whether Q is the only or the best theory for explanation of incompleteness and undecidability phenomena. In connection with this question, A. Grzegorczyk in [Grz05] proposed to study the theory TC, the theory of concatenation. Instead of numbers that can be added and multiplied, in TC one has strings that can be concatenated, and there are two irreducible (single-letter) strings a and b. Some ideas behind formulation of axioms of TC go back to Quine [Qui46] and Tarski. Grzegorczyk’s motivations to study the theory TC are philosophical and are explained in Introduction and in the beginning of Section 8 of [Grz05]. Speaking briefly, when reasoning, computing, This work is a part of the research plan MSM 0021620839 that is financed by the Ministry of Education of the Czech Republic. The author also acknowledges support by a Fulbright Scholarship at the Department of Philosophy, University of Notre Dame in Notre Dame, IN. Charles University, Prague, vitezslavdotsvejdaratcunidotcz, http://www1.cuni.cz/ svejdar/. Palachovo nam. 2, 116 38 Praha 1, Czech Republic.