Abstract
This article is essentially a part of my thesis for the degree DrSc (Doctor of Sciences). Therefore it mainly surveys my articles [42, 43, 44, 29, 30, 45, 23], and it is structured according to the requirements for such theses. I made only minor changes in the original text and added a few further references. Since Gödel’s main achievement concerns the problem of consistency and some of the problems that I am going to describe had been considered by him, I think that it is appropriate to publish this article in Gödel Society.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Miklos Ajtai. The complexity of the pigeonhole principle. In Proceedings of the 29th Annual IEEE Symposium on Foundations of Computer Science, pages 346–355, 1988.
Miklos Ajtai. The independence of the modulo p counting principles. In Proc. 26th ACM Symp. on Theory of Computing,pages 402–411, 1994.
Mathias Baaz and Pavel Pudlák. Kreisel’s conjecture for Lat. In Peter Clote and Jan Krajícek, editors, Arithmetic Proof Theory and Computational Complexity, pages 30–39. Oxford Univ. Press, 1993.
Paul Beame, Russell Impagliazzo, Jan Krajícek, Toniann Pitassi, Pavel Pudlák, and Alan Woods. Exponential lower bounds for the pigeonhole principle. In Proceedings of the 24th Annual ACM Symposium on Theory of Computing, pages 200–221, 1992.
A. Bezboruah and J. Shepherdson. Gödel’s second incompleteness theorem for Q. J. Symbolic Logic, pages 503–512, 1976.
Samuel R. Buss. Bounded Arithmetic. Bibliopolis, 1986. Revision of 1985 Princeton University Ph.D. thesis.
Samuel R. Buss and Aleksandar Ignjatović. Unprovability of consistency statements in fragments of bounded arithmetic. In preparation, 1993.
R. Carnap. Logische Syntax der Sprache. Springer-Verlag, 1934.
Stephen A. Cook. Feasibly constructive proofs and the propositional calculus. In Proceedings of the 7th Annual ACM Symposium on Theory of Computing,pages 83–97, 1975.
Martin Dowd. Propositional Representation of Arithmetic Proofs. PhD thesis, University of Toronto, 1979.
A.G. Dragalin. Correctness of inconsistent theories with notions of feasibility. volume 108 of Lecture Notes in Comp. Sci.,pages 58–79, 1985.
A. Esenine-Volpin. Le programme ultra-intuitionniste des fondements des mathematiques. In Infinitistic Methods, Proceedings of the Symposium on Foundations of Mathematics, pages 201–223. PWN, Warsaw, 1961.
J. Ferrante and Ch. W. Rackoff. The Computational Complexity of Logical Theories. LNM 718. Springer-Verlag, 1979.
H. Friedman. On the consistency, completeness, and correctness problems. Ohio State Univ., unpublished, 1979.
H. Friedman. Translatability and relative consistency II. Ohio State Univ., unpublished, 1979.
Yu.V. Gavrilenko. Monotone theories of feasible numbers. Doklady Akademii Nauk SSSR, 276(1):18–22, 1984.
Gerhard Gentzen. Die Widerspruchsfreiheit der reinen Zahlentheorie. Mathematische Annalen, 112:493–565, 1936. English translation in Gerhard Gentzen, Collected Papers of Gerhard Gentzen, Editted by M. E. Szabo, North-Holland, 1969, pp. 132–213.
Gerhard Gentzen. Collected Papers of Gerhard Gentzen. North-Holland, 1969. Editted by M. E. Szabo.
K. Gödel. Uber formal unentscheidbare Sätze der Principia Mathematica und vewandter Systeme I. Monatshefte Math. Phys., 38:173–198, 1931.
K. Gödel. Uber die Länge von Beweisen. Ergebnisse eines Mathematischen Kolloquiums,pages 23–24, 1936. English translation in Kurt Gödel: Collected Works, Volume 1, pages 396–399, Oxford University Press, 1986.
P. Hájek. On interpretability in set theories II. Commentatione Math. Univ. Carol., 13:445–455, 1972.
Petr Häjek. On a new notion of partial conservativity. In Logic Colloquium ‘83, pages 217–232. Springer-Verlag, 1983.
Petr Häjek, Franco Montagna, and Pavel Pudlák. Abbreviating proofs using meta-mathematical rules. In Peter Clote and Jan Krajícek, editors, Arithmetic Proof Theory and Computational Complexity, pages 197–221. Oxford Univ. Press, 1993.
Petr Häjek and Pavel Pudlák. Metamathematics of First-order Arithmetic. Springer-Verlag, 1993.
D. Hilbert. Die Grundlagen der Mathematik, volume 5 of Hamburger Mathematische Einzelschriften. Teubner, 1934. A lecture presented in Hamburg in July 1927.
A. Ignjatovié. Fragments of First and Second Order Arithmetic and Length of Proofs. PhD thesis, University of California at Berkeley, 1990.
Jan Krajíček. A note on proofs of falsehood. Archiv f. Math. Logic u. Grundlagen d. Math., 26:169–179, 1987.
Jan Krajíček. Lower bounds to the size of constant-depth propositional proofs. Journal of Symbolic Logic, 59(1):73–86, 1994.
Jan Krajícek and Pavel Pudlâk. On the structure of initial segments of models of arithmetic. Archive for Mathematical Logic, 28:91–98, 1989.
Jan Krajícek and Pavel Pudlâk. Propositional proof systems, the consistency of first-order theories and the complexity of computations. Journal of Symbolic Logic, 54:1063–1079, 1989.
Jan Krajícek and Pavel Pudlâk. Quantified propositional calculi and fragments of bounded arithmetic. Zeitschrift für Mathematische Logik and Grundlagen der Mathematik,36:29–46, 1990.
Jan Krajícek, Pavel Pudlâk, and Gaisi Takeuti. Bounded arithmetic and the polynomial hierarchy. Annals of Pure and Applied Logic, 52:143–154, 1991.
Jan Krajíček and Gaisi Takeuti. On induction-free provability. Annals of Math. and Artificial Intelligence, 6:107–126, 1992.
R. Montague. Theories incomparable with respect to relative interpretability. Journal of Symbolic Logic, 27:195–211, 1962.
V. P. Orevkov. Correctness of short proofs in theory with notions of feasibility. volume 417 of Lecture Notes in Comp. Sci.,pages 242–245, 1990.
S. Orey. Relative interpretations. Journal of Symbolic Logic, 24:281–282, 1959.
S. Orey. Relative interpretations. Zeitschrift für Mathematische Logik and Grundlagen der Mathematik, 7:146–153, 1961.
R. Parikh. Existence and feasibility in arithmetic. J. Symbolic Logic, 36:494–508, 1971.
J. B. Paris and C. Dimitracopoulos. Truth definitions for formulas. In Logic et algoritmic, L ‘enscignment Mathematique No 30, pages 318–329, 1982.
J. B. Paris and C. Dimitracopoulos. A note on the undefinability of cuts. J. Symbolic Logic, 48:564–569, 1983.
J.B. Paris and L. Harrington. A mathematical incompleteness in Peano Arithmetic. In Handbook of Mathematical Logic, pages 1133–1142. North-Holland, 1977.
Pavel Pudlák. Cuts, consistency statements and interpretation. Journal of Symbolic Logic, 50:423–441, 1985.
Pavel Pudlák. On the lengths of proofs of finitistic consistency statements in first order theories. In Logic Colloquium ‘84, pages 165–196. North-Holland, 1986.
Pavel Pudlák. Improved bounds to the lengths of proofs of finitistic consistency statements. In Logic and Combinatorics, volume 65 of Contemporary Mathematics, pages 309–331. American Mathematical Society, 1987.
Pavel Pudlák. A note on bounded arithmetic. Fundamenta Mathematicae, 136:85–89, 1990.
C. Smoriński. Nonstandard models and related developments. In Harvey Fried-man’s Research on the Foundations of Mathematics, pages 179–229. North Holland, 1985.
R.M. Solovay. Injecting inconsistencies into models of PA. Annals of Pure and Applied Logic, 44:101–132, 1989.
R.M. Solovay. Upper bounds on the speedup of GB over ZF. preprint, 1990.
R. Statman. Proof search and speed-up in the predicate calculus. Ann. Math. Logic, 15:225–287, 1978.
V. Švejdar. Modal analysis of generalized Rosser sentences. Journ. of Symb. Logic, 48:986–999, 1983.
G. Takeuti. Some relations among systems for bounded arithmetic. Annals of Pure and Applied Logic, 39:75–104, 1988.
O.V. Verbitsky. Optimal algorithms for coNP-sets and the problem EXP=?NEXP. Matematicheskie zametki, 50(2):37–46, 1991. Eglish translation in: Math. Notes 50,1–2, (1991), pp. 798–801.
L.Ch. Verbrugge. Efficient Mathematics. PhD thesis, Universiteit van Amsterdam, 1993.
A. Visser. The unprovability of small consistency. Archive for Math. Logic, 32:275–298, 1993.
A. Wilkie. On sentences interpretable in systems of arithmetic. In Logic Colloquium ‘84, pages 329–342. North-Holland, 1986.
A.J. Wilkie and J.B. Paris. On the schema of induction for bounded arithmetical formulas. Annals of Pure and Applied Logic, 35:261–302, 1987.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1996 Springer-Verlag/Wien
About this paper
Cite this paper
Pudlák, P. (1996). On the lengths of proofs of consistency. In: Collegium Logicum. Collegium Logicum, vol 2. Springer, Vienna. https://doi.org/10.1007/978-3-7091-9461-4_5
Download citation
DOI: https://doi.org/10.1007/978-3-7091-9461-4_5
Publisher Name: Springer, Vienna
Print ISBN: 978-3-211-82796-3
Online ISBN: 978-3-7091-9461-4
eBook Packages: Springer Book Archive