Undecidability Results on Two-Variable Logics

It is a classical result of Mortimer that L 2 , rst-order logic with two variables, is decidable for satissability. 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 xed-point operations weak access to cardinalities, through the HH artig (or equicardinality) quantiier a choice construct known as Hilbert's "-operator. In fact all these extensions of L 2 prove to be undecidable both for satissability, and for satissability in nite models. Moreover most of them are hard for 1 1 , the rst level of the analytical hierachy, and thus have a much higher degree of undecidability than rst-order logic.