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Arithmetic on semigroups

Journal of Symbolic Logic 74 (1):265-278 (2009)

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  1. On Interpretability in the Theory of Concatenation.Vítězslav Švejdar - 2009 - Notre Dame Journal of Formal Logic 50 (1):87-95.
    We prove that a variant of Robinson arithmetic $\mathsf{Q}$ with nontotal operations is interpretable in the theory of concatenation $\mathsf{TC}$ introduced by A. Grzegorczyk. Since $\mathsf{Q}$ is known to be interpretable in that nontotal variant, our result gives a positive answer to the problem whether $\mathsf{Q}$ is interpretable in $\mathsf{TC}$. An immediate consequence is essential undecidability of $\mathsf{TC}$.
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  • Variants of Robinson's essentially undecidable theoryR.James P. Jones & John C. Shepherdson - 1983 - Archive for Mathematical Logic 23 (1):61-64.
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  • Undecidability without Arithmetization.Andrzej Grzegorczyk - 2005 - Studia Logica 79 (2):163-230.
    In the present paper the well-known Gödels – Churchs argument concerning the undecidability of logic (of the first order functional calculus) is exhibited in a way which seems to be philosophically interestingfi The natural numbers are not used. (Neither Chinese Theorem nor other specifically mathematical tricks are applied.) Only elementary logic and very simple set-theoretical constructions are put into the proof. Instead of the arithmetization I use the theory of concatenation (formalized by Alfred Tarski). This theory proves to be an (...)
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