Abstract
We firstly survey several forms of Herbrand's theorem. What is commonly called “Herbrand's theorem” in many textbooks is actually a very simple form of Herbrand's theorem which applies only to ℬ∃-formulas; but the original statement of Herbrand's theorem applied to arbitrary first-order formulas. We give a direct proof, based on cut-elimination, of what is essentially Herbrand's original theorem. The “nocounterexample theorems” recently used in bounded and Peano arithmetic are immediate corollaries of this form of Herbrand's theorem. Secondly, we discuss the results proved in Herbrand's 1930 dissertation.
Supported in part by NSF grant DMS-9205181
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
S. R. Buss, An introduction to proof theory. Typeset manuscript, to appear in Handbook of Proof Theory, 199?
-, Relating the bounded arithmetic and polynomial-time hierarchies. To appear, Annals of Pure and Applied Logic, 199?
B. Dreben and S. Aanderaa, Herbrand analyzing functions, Bulletin of the American Mathematical Society, 70 (1964), pp. 697–698.
B. Dreben, P. Andrews, and S. Aanderaa, False lemmas in Herbrand, Bulletin of the American Mathematical Society, 69 (1963), pp. 699–706.
B. Dreben and J. Denton, A supplement to Herbrand, Journal of Symbolic Logic, 31 (1966), pp. 393–398.
W. D. Goldfarb, Herbrand's theorem and the incompleteness of arithmetic, Iyyun, A Jerusalem Philosophical Quarterly, 39 (1990), pp. 45–64.
-Herbrand's error and Gödel's correction, Modern Logic, 3 (1993), pp. 103–118.
J. Herbrand, Recherches sur la théorie de la démonstration, PhD thesis, University of Paris, 1930.
-Investigations in proof theory: The properties of true propositions, in From Frege to Gödel: A Source Book in Mathematical Logic, 1978–1931, J. van Heijenoort, ed., Harvard University Press, Cambridge, Massachusetts, 1967, pp. 525–581. Translation of chapter 5 of [8], with commentary and notes, by J. van Heijenoort and B. Dreben.
-Écrits logique, Presses Universitaires de France, Paris, 1968. Ed. by J. van Heijenoort.
-Logical Writings, D. Reidel, Dordrecht-Holland, 1971. Ed. by W. Goldfarb, Translation of [10].
J. Krajíček, P. Pudlák, and G. Takeuti, Bounded arithmetic and the polynomial hierarchy, Annals of Pure and Applied Logic, 52 (1991), pp. 143–153.
G. Kreisel, On the interpretation of non-finitist proofs-part I, Journal of Symbolic Logic, 16 (1951), pp. 241–267.
-On the interpretation of non-finitist proofs, part II. interpretation of number theory, applications, Journal of Symbolic Logic, 17 (1952), pp. 43–58.
M. S. Paterson and M. N. Wegman, Linear unification, J. Comput. System Sci., 16 (1978), pp. 158–167.
J. A. Robinson, A machine-oriented logic based on the resolution principle, J. Assoc. Comput. Mach., 12 (1965), pp. 23–41.
G. Takeuti, Proof Theory, North-Holland, Amsterdam, 2nd ed., 1987.
D. Zambella, Notes on polynomially bounded arithmetic. To appear in J. Symb. Logic.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1995 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Buss, S.R. (1995). On Herbrand's theorem. In: Leivant, D. (eds) Logic and Computational Complexity. LCC 1994. Lecture Notes in Computer Science, vol 960. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60178-3_85
Download citation
DOI: https://doi.org/10.1007/3-540-60178-3_85
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-60178-4
Online ISBN: 978-3-540-44720-7
eBook Packages: Springer Book Archive