Skip to main content
    • Aa
    • Aa

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

  • William Craig (a1)

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].

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[5] G. Gentzen , Untersuchungen über das logische Schliessen, Mathematische Zeitschrift, vol. 39 (1934–1935), pp. 176–210, 405431.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 0
Total number of PDF views: 3 *
Loading metrics...

Abstract views

Total abstract views: 146 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 28th June 2017. This data will be updated every 24 hours.