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Random models and the Gödel case of the decision problem

  • Yuri Gurevich (a1) and Saharon Shelah (a2)

Abstract

In a paper of 1933 Gödel proved that every satisfiable first-order ∀2∃* sentence has a finite model. Actually he constructed a finite model in an ingenious and sophisticated way. In this paper we use a simple and straightforward probabilistic argument to establish existence of a finite model of an arbitrary satisfiable ∀2∃* sentence.

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[1]Dreben, Burton S. and Goldfarb, Warren D., The decision problem: Solvable classes of quantificational formulas, Addison-Wesley, Reading, Mass., 1979.
[2]Fagin, Ronald, Probabilities on finite models, this Journal, vol. 41 (1976), pp. 5057.
[3]Gödel, Kurt, Zum Entscheidungsproblem des logischen Funktionenkalküls, Monatshefte für Mathematik und Physik, vol. 40 (1933), pp. 433443.
[4]Goldfarb, Warren D., On the Gödel class with identity, this Journal, vol. 46 (1981), pp. 354364.
[5]Goldfarb, Warren D., Gurevich, Yuri and Shelah, Saharon, On the Gödel class with identity (in preparation).
[6]Kalmar, Laszlo, Über die Erfüllbarkeit derjenigen Zahlausdrucke, welche in der Normalform zwei benachtbarte Allzeichen enthalten, Mathematische Annalen, vol. 108 (1933), pp. 466484.
[7]Lewis, Harry R., Unsohable classes of quanlificational formulas, Addison-Wesley, Reading, Mass., 1979.
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[10]Schütte, Kurt, Über die Erfüllbarkeit einer Klasse von logischen Formeln, Mathematische Annalen, vol. 110 (1934), pp. 161194.

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