A decidable subclass of the minimal Gödel class with identity

Warren D. Goldfarb; Yuri Gurevich; Saharon Shelah

doi:10.2307/2274275

Published online by Cambridge University Press: 12 March 2014

Warren D. Goldfarb ,

Yuri Gurevich and

Saharon Shelah

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Warren D. Goldfarb

Affiliation:

Department of Philosophy, Harvard University, Cambridge, Massachusetts 02138

Yuri Gurevich

Affiliation:

Department of Computer and Communication Sciences, University of Michigan, Ann Arbor, Michigan 48104

Saharon Shelah

Affiliation:

Department of Mathematics, Hebrew University, Jerusalem, Israel

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The minimal Gödel class with identity (MGCI) is the class of closed, prenex quantificational formulas whose prefixes have the form ∀x1∀x2∃x3 and whose matrices contain arbitrary predicate letters and the identity sign “=”, but contain no function signs or individual constants. The MGCI was shown undecidable (for satisfiability) in 1983 [Go2]; this both refutes a claim of Gödel's [Gö, p. 443] and settles the decision problem for all prefix-classes of quantification theory with identity.

In this paper, we show the decidability of a natural subclass of the MGCI. The formulas in this subclass can be thought of as exploiting only half of the power of the existential quantifier. That is, since an MGCI formula has prefix ∀x1∀x2∃x3, in general its truth in a model requires for any elements a and b, the existence of both a witness for and a witness for . The formulas we consider demand less: they require, for any elements a and b, a witness for the unordered pair {a, b}, that is, a witness either for or for .

References

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[Go2]Goldfarb, W. D., The unsolvability of the Godel class with identity, this Journal, vol. 49 (1984), pp. 1237–1252.Google Scholar

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