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Undecidability results on two-variable logics

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Abstract.

It is a classical result of Mortimer that \(L^2\), first-order logic with two variables, is decidable for satisfiability. We show that going beyond \(L^2\) by adding any one of the following leads to an undecidable logic:– very weak forms of recursion, viz.¶(i) transitive closure operations¶(ii) (restricted) monadic fixed-point operations¶– weak access to cardinalities, through the Härtig (or equicardinality) quantifier¶– a choice construct known as Hilbert's \(\epsilon\)-operator.

In fact all these extensions of \(L^2\) prove to be undecidable both for satisfiability, and for satisfiability in finite structures. Moreover most of them are hard for \(\Sigma^1_1\), the first level of the analytical hierachy, and thus have a much higher degree of undecidability than first-order logic.

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Received: 13 July 1996

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Grädel, E., Otto, M. & Rosen, E. Undecidability results on two-variable logics. Arch Math Logic 38, 313–354 (1999). https://doi.org/10.1007/s001530050130

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  • DOI: https://doi.org/10.1007/s001530050130

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