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The unsolvability of the Gödel class with identity

Published online by Cambridge University Press:  12 March 2014

Warren D. Goldfarb
Warren D. Goldfarb*
Affiliation:
Harvard University Cambridge, Massachusetts 02138
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Extract

The Gödel class with identity (GCI) is the class of closed, prenex quantificational formulas whose prefixes have the form ∀∀∃ … ∃ and whose matrices contain arbitrary predicate letters and the identity sign “ = ”, but do not contain function signs or individual constants. The ∀∀∃ … ∃ class without identity was shown solvable over fifty years ago ([4], [12], [17]); slightly later, that class was shown to possess the stronger property of finite controllability ([5], [18]). (A class of formulas is solvable iff it is decidable for satisfiability; it is finitely controllable iff every satisfiable formula in it has a finite model.) At the end of [5], Gödel claims that the finite controllability of the GCI can be shown “by the same method” as he employed to show this for the class without identity. This claim has been questioned for nearly twenty years; in §1 below we give a brief history of investigations into it. The major result of this paper shows Gödel to have been mistaken: the GCI is unsolvable. §2 contains the basic construction, which yields a satisfiable formula in the GCI that lacks finite models. This formula may easily be exploited to encode undecidable problems into the GCI.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 49 , Issue 4 , December 1984 , pp. 1237 - 1252
Copyright
Copyright © Association for Symbolic Logic 1984

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