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Three uses of the Herbrand-Gentzen theorem in relating model theory and proof theory

Published online by Cambridge University Press:  12 March 2014

William Craig*
Affiliation:
The Pennsylvania State University

Extract

One task of metamathematics is to relate suggestive but nonelementary modeltheoretic concepts to more elementary proof-theoretic concepts, thereby opening up modeltheoretic problems to proof-theoretic methods of attack. Herbrand's Theorem (see [8] or also [9], vol. 2) or Gentzen's Extended Hauptsatz (see [5] or also [10]) was first used along these lines by Beth [1]. Using a modified version he showed that for all first-order systems a certain modeltheoretic notion of definability coincides with a certain proof theoretic notion. In the present paper the Herbrand-Gentzen Theorem will be applied to generalize Beth's results from primitive predicate symbols to arbitrary formulas and terms.

This may be interpreted as showing that (apart from some relatively minor exceptions which will be made apparent below) the expressive power of each first-order system is rounded out, or the system is functionally complete, in the following sense: Any functional relationship which obtains between concepts that are expressible in the system is itself expressible and provable in the system.

A second application is concerned with the hierarchy of second-order formulas. A certain relationship is shown to hold between first-order formulas and those second-order formulas which are of the form (∃T1)…(∃Tk)A or (T1)…(Tk)A with A being a first-order formula. Modeltheoretically this can be regarded as a relationship between the class AC and the class PC of sets of models, investigated by Tarski in [12] and [13].

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1957

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References

[1]Beth, E. W., On Padoa's method in the theory of definitions, Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, ser. A, vol. 56 (also Indagationes mathematicae, vol. 15) (1953), pp. 330339.Google Scholar
[2]Birkhoff, Garrett, Lattice theory, American Mathematical Society Colloquium Publications, vol. 25 (1940).Google Scholar
[3]Büchi, J. R. and Craig, W., Notes on the family PC of sets of models, abstract, this Journal, vol. 21 (1956), pp. 222223.Google Scholar
[4]Craig, W., Linear reasoning. A new form of the Herbrand-Gentzen Theorem, this Journal, vol 22 (1957), pp 250268.Google Scholar
[5]Gentzen, G., Untersuchungen über das logische Schliessen, Mathematische Zeitschrift, vol. 39 (1934–1935), pp. 176–210, 405431.CrossRefGoogle Scholar
[6]Hailperin, T., Remarks on identity and description in first-order systems, this Journal, vol. 19 (1954), pp. 1420.Google Scholar
[7]Henkin, L., The completeness of the first-order functional calculus, this Journal, vol. 14 (1949), pp. 159166.Google Scholar
[8]Herbrand, J., Recherches sur la théorie de la démonstration, Travaux de la Société des Sciences et Lettres de Varsovie, Classe III sciences mathématiques et physiques, no. 33 (1930).Google Scholar
[9]Hilbert, D. and Bernays, P., Grundlagen der Mathematik, Berlin (Julius Springer), vol. 1 (1934) and vol. 2 (1939).Google Scholar
[10]Kleene, S. C., Introduction to metamathematics, Amsterdam (North Holland), Groningen (Noordhoff), New York and Toronto (Van Nostrand), 1952.Google Scholar
[11]Robinson, A., A result on consistency and its application to the theory of definition, Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, ser. A, vol. 59 (also Indagationes mathematicae, vol. 18) (1956), pp. 4758.Google Scholar
[12]Tarski, A., Some notions and methods on the borderline of algebra and metamathematics, Proceedings of the International Congress of Mathematicians, 1950, vol. 1, pp. 705720.Google Scholar
[13]Tarski, A., Contributions to the theory of models, I and II, Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings, ser. A, vol. 57 (also Indagationes mathematicae, vol. 16) (1954), pp. 572588.Google Scholar
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