# Completeness Theorem for First-Order Logic

• Shashi Mohan Srivastava
Chapter
Part of the Universitext book series (UTX)

## Abstract

In Chap. 1, we described what a first-order language is and what its terms and formulas are. We fixed a first-order language L. In Chap. 2, we described the semantics of first-order languages. In Chap. 3, we considered a simpler form of logic – propositional logic, defined what a proof is in that logic, and proved its completeness theorem. In this chapter we shall define proof in a first-order theory and prove the corresponding completeness theorem. The result for countable theories was first proved by Gödel in 1930. The result in its complete generality was first observed by Malcev in 1936. The proof given below is due to Leo Henkin.

## Keywords

Propositional Logic Completeness Theorem Canonical Structure Constant Symbol Truth Valuation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Ax, J.: The elementary theory of finite fields. Ann. Math. 88, 103–115 (1968)
2. 2.
Bochnak, J., Coste, M., Roy, M-F.: Real Algebraic Geometry, vol. 36, A Series of Modern Surveys in Mathematics. Springer, New York (1998)
3. 3.
Chang, C.C., Keisler, H.J.: Model Theory, 3rd edn. North-Holland, London (1990)
4. 4.
Flath, D., Wagon, S.: How to pick out integers in the rationals: An application of number theory to logic. Am. Math. Mon. 98, 812–823 (1991)
5. 5.
Hinman, P.: Fundamentals of Mathematical Logic. A. K. Peters (2005)
6. 6.
Hofstadter, D.R.: Gödel, Escher, Bach: An Eternal Golden Braid. Vintage Books, New York (1989)Google Scholar
7. 7.
Hrushovski, E.: The Mordell–Lang conjecture for function fields. J. Am. Math. Soc. 9(3), 667–690 (1996)
8. 8.
Jech, T.: Set Theory, Springer Monographs in Mathematics, 3rd edn. Springer, New York (2002)Google Scholar
9. 9.
Kunen, K.: Set Theory: An Introduction to Independence Proofs. North-Holland, Amsterdam (1980)
10. 10.
11. 11.
Marker, D.: Model Theory: An Introduction, GTM 217. Springer, New York (2002)Google Scholar
12. 12.
Rogers, H.J.: Theory of Recursive Functions and Effective Computability. McGraw-Hill, New York (1967)
13. 13.
Penrose, R.: The Emperor’s New Mind. Oxford University Press, Oxford (1990)Google Scholar
14. 14.
Pila, J.: O-minimality and André-Oort conjecture for n. Ann. Math. (2) 172(3), 1779–1840 (2011)Google Scholar
15. 15.
Pila, J., Zannier, U.: Rational points in periodic analytic sets and the Manin-Mumford conjecture, Atti. Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei(9). Mat. Appl. 19(2), 149–162 (2008)Google Scholar
16. 16.
Shoenfield, J.R.: Mathematical Logic. A. K. Peters (2001)
17. 17.
Srivastava, S.M.: A Course on Borel Sets, GTM 180. Springer, New York (1998)
18. 18.
Swan, R.G.: Tarski’s principle and the elimination of quantifiers (preprint)Google Scholar 