Complete logical Calculi

  • Wolfgang Rautenberg
Part of the Universitext book series (UTX)


Our first goal is to characterize the consequence relation in a first-order language by means of a calculus similar to that of propositional logic. That this goal is attainable at all was shown for the first time by Gdel in [Gö1]. The original version of Gdel’s theorem refers to the axiomatization of tautologies only and does not immediately imply the compactness theorem of first-order logic; but a more general formulation of completeness in 3.2 does. The importance of the compactness theorem for mathematical applications was first revealed in 1936 by A. Malcev, see [Ma].

The characterizability of logical consequence by means of a calculus (the content of the completeness theorem) is a crucial result in mathematical logic with far-reaching applications. In spite of its metalogical origin, the completeness theorem is essentially a mathematical theorem. It satisfactorily explains the phenomenon of the well-definedness of logical deductive methods in mathematics. To seek any additional, possibly unknown methods or rules of inference would be like looking for perpetual motion in physics. Of course, this insight does not affect the development of new ideas in solving open questions. We will say somewhat more regarding the metamathematical aspect of the theorem and its applications, as well as the use of the model construction connected with its proof in a partly descriptive manner, in 3.3, 3.4, and 3.5.


Propositional Logic Axiom System Compactness Theorem Natural Deduction Completeness Theorem 
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  1. Ma.
    A. I. Mal’​cev, The Metamathematics of Algebraic Systems, North-Holland 1971.Google Scholar
  2. Me.
    E. Mendelson, Introduction to Mathematical Logic, Princeton 1964, 4⟨{ th}⟩ ed. Chapman & Hall 1997.Google Scholar
  3. Ge.
    G. Gentzen, The Collected Papers of Gerhard Gentzen (editor M. E. Szabo), North-Holland 1969.Google Scholar
  4. Po.
    W. Pohlers, Proof Theory, An Introduction, Lecture Notes in Mathematics 1407, Springer 1989.Google Scholar
  5. Kra.
    J. Krajíček, Bounded Arithmetic, Propositional Logic, and Complexity Theory, Cambridge Univ. Press 1995.Google Scholar
  6. Ry.
    C. Ryll-Nardzewki, The role of the axiom of induction in elemenary arithmetic, Fund. Math. 39 (1952), 239–263.Google Scholar
  7. HP.
    P. Hájek, P. Pudlák, Metamathematics of First-Order Arithmetic, Springer 1993.Google Scholar
  8. BGG.
    E. Börger, E. Grädel, Y. Gurevich, The Classical Decision Problem, Springer 1997.Google Scholar
  9. Id.
    P. Idziak, A characterization of finitely decidable congruence modular varieties, Trans. Amer. Math. Soc. 349 (1997), 903–934.Google Scholar
  10. Sz.
    W. Szmielew, Elementary properties of abelian groups, Fund. Math. 41 (1954), 203–271.Google Scholar
  11. Bi.
    G. Birkhoff, On the structure of abstract algebras, Proceedings of the Cambridge Philosophical Society 31 (1935), 433–454.Google Scholar
  12. Se.
    A. Selman, Completeness of calculi for axiomatically defined classes of algebras, Algebra Universalis 2 (1972), 20–32.Google Scholar
  13. Mo.
    D. Monk, Mathematical Logic, Springer 1976.Google Scholar
  14. EFT.
    H.-D. Ebbinghaus, J. Flum, W. Thomas, Mathematical Logic, New York 1984, 2⟨{ nd}⟩ ed. Springer 1994.Google Scholar
  15. CK.
    C. C. Chang, H. J. Keisler, Model Theory, Amsterdam 1973, 3⟨{ rd}⟩ ed. North-Holland 1990.Google Scholar
  16. Gö1.
    K. Gödel, Die Vollständigkeit der Axiome des logischen Funktionenkalküls, Monatshefte f. Math. u. Physik 37 (1930), 349–360, also in [Gö3, Vol. I, 102–123], [Hei, 582–591].Google Scholar
  17. Ta4.
    _________ , Logic, Semantics, Metamathematics (editor J. Corcoran), Oxford 1956, 2⟨{ nd}⟩ ed. Hackett 1983.Google Scholar
  18. TMR.
    A. Tarski, A. Mostowski, R. M. Robinson, Undecidable Theories, North-Holland 1953.Google Scholar
  19. Ra5.
    _________ , Messen und Zhlen, Eine einfache Konstruktion der reellen Zahlen, Heldermann 2007.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Fachbereich Mathematik und InformatikBerlinGermany

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