Abstract
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.
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Notes
- 1.
- 2.
Whenever \(\mathcal{A}\) is embeddable into \(\mathcal{B}\) there is a structure \(\mathcal{B}'\) isomorphic to \(\mathcal{B}\) such that \(\mathcal{A}\subseteq \mathcal{B}'\). The domain B′ arises from B by interchanging the images of the elements of \(\mathcal{A}\) with their originals.
- 3.
Here we use the axiom of choice, since for every M i some enumeration is chosen. It can be shown that without the axiom of choice the proof cannot be carried out.
- 4.
A frequently used synonym for the domain of a ZFC-model, mostly denoted by V, and ‘for all sets a’ is then often expressed as ‘for all a ∈ V ’.
- 5.
In 6.6 the incompleteness of ZFC and all its axiomatic extensions will be proved.
- 6.
The elementary absolute (plane) geometry T has precisely two completions, Euclidean and non-Euclidean (or hyperbolic) geometry. Both are axiomatizable, hence decidable. Completeness follows in either case from the completeness of the elementary theory of real numbers, Theorem 5.5.5. Thus, absolute geometry is decidable as well. Further applications can be found in 5.2.
- 7.
Conversely, if a class K has these three properties then K is a variety. This is Birkhoff’s HSP theorem, a basic theorem of universal algebra; see e.g. [Mo].
References
A. I. Mal’cev, The Metamathematics of Algebraic Systems, North-Holland 1971.
E. Mendelson, Introduction to Mathematical Logic, Princeton 1964, 4⟨{ th}⟩ ed. Chapman & Hall 1997.
G. Gentzen, The Collected Papers of Gerhard Gentzen (editor M. E. Szabo), North-Holland 1969.
W. Pohlers, Proof Theory, An Introduction, Lecture Notes in Mathematics 1407, Springer 1989.
J. Krajíček, Bounded Arithmetic, Propositional Logic, and Complexity Theory, Cambridge Univ. Press 1995.
C. Ryll-Nardzewki, The role of the axiom of induction in elemenary arithmetic, Fund. Math. 39 (1952), 239–263.
P. Hájek, P. Pudlák, Metamathematics of First-Order Arithmetic, Springer 1993.
E. Börger, E. Grädel, Y. Gurevich, The Classical Decision Problem, Springer 1997.
P. Idziak, A characterization of finitely decidable congruence modular varieties, Trans. Amer. Math. Soc. 349 (1997), 903–934.
W. Szmielew, Elementary properties of abelian groups, Fund. Math. 41 (1954), 203–271.
G. Birkhoff, On the structure of abstract algebras, Proceedings of the Cambridge Philosophical Society 31 (1935), 433–454.
A. Selman, Completeness of calculi for axiomatically defined classes of algebras, Algebra Universalis 2 (1972), 20–32.
D. Monk, Mathematical Logic, Springer 1976.
H.-D. Ebbinghaus, J. Flum, W. Thomas, Mathematical Logic, New York 1984, 2⟨{ nd}⟩ ed. Springer 1994.
C. C. Chang, H. J. Keisler, Model Theory, Amsterdam 1973, 3⟨{ rd}⟩ ed. North-Holland 1990.
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].
_________ , Logic, Semantics, Metamathematics (editor J. Corcoran), Oxford 1956, 2⟨{ nd}⟩ ed. Hackett 1983.
A. Tarski, A. Mostowski, R. M. Robinson, Undecidable Theories, North-Holland 1953.
_________ , Messen und Zhlen, Eine einfache Konstruktion der reellen Zahlen, Heldermann 2007.
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Rautenberg, W. (2010). Complete logical Calculi. In: A Concise Introduction to Mathematical Logic. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-1221-3_3
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