On the Gödel class with identity

Abstract

The Gödel Class is the class of prenex formulas of pure quantification theory whose prefixes have the form ∀y1∀y2∃x1 … ∃xn. The Gödel Class with Identity, or GCI, is the corresponding class of formulas of quantification theory extended by inclusion of the identity-sign “ = ”. Although the Gödel Class has long been kndwn to be solvable, the decision problem for the Gödel Class with Identity is open. In this paper we prove that there is no primitive recursive decision procedure for the GCI, or, indeed, for the subclass of the GCI containing just those formulas with prefixes ∀y1∀y2∃x.Throughout this paper we take quantification theory to include, aside from logical signs, infinitely many k-place predicate letters for each k > 0, but no function signs or constants. Moreover, by “prenex formula” we include only those without free variables. A decision procedure for a class of formulas is a recursive function that carries a formula in the class to 0 if the formula is satisfiable and to 1 if not. A class is solvable iff there exists a decision procedure for it. A class is finitely controllable iff every satisfiable formula in the class has a finite model. Since we speak only of effectively specified classes, finite controllability implies solvability (but not conversely).The GCI has a curious history. Gödel showed the Gödel Class (without identity) solvable in 1932 [4] and finitely controllable in 1933 [5].