An Open Formalism against Incompleteness
Francesc Tomàs
Source: Notre Dame J. Formal Logic Volume 40, Number 2
(1999), 207-226.
Abstract
An open formalism for arithmetic is presented based on first-order logic supplemented by a very strictly controlled constructive form of the omega-rule. This formalism (which contains Peano Arithmetic) is proved (nonconstructively, of course) to be complete. Besides this main formalism, two other complete open formalisms are presented, in which the only inference rule is modus ponens. Any closure of any theorem of the main formalism is a theorem of each of these other two. This fact is proved constructively for the stronger of them and nonconstructively for the weaker one. There is, though, an interesting counterpart: the consistency of the weaker formalism can be proved finitarily.
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Permanent link to this document: http://projecteuclid.org/euclid.ndjfl/1038949537
Mathematical Reviews number (MathSciNet): MR1816889
Digital Object Identifier: doi:10.1305/ndjfl/1038949537
Zentralblatt MATH identifier: 0972.03058
References
[1] Boolos, G., The Logic of Provability, Cambridge University Press, Cambridge, 1973.
Mathematical Reviews (MathSciNet): MR95c:03038
Zentralblatt MATH: 0891.03004
[2] Church, A., ``The constructive second number class,'' Bulletin of the American Mathematical Society, vol. 44 (1938), pp. 224--32.
Zentralblatt MATH: 0018.33803
Mathematical Reviews (MathSciNet): MR1563715
Digital Object Identifier: doi:10.1090/S0002-9904-1938-06720-1
Project Euclid: euclid.bams/1183500400
[3] Davis, M., ``Hilbert's tenth problem is unsolvable,'' American Mathematical Monthly, vol. 80 (1973), pp. 233--69.
Mathematical Reviews (MathSciNet): MR47:6465
Zentralblatt MATH: 0277.02008
Digital Object Identifier: doi:10.2307/2318447
JSTOR: links.jstor.org
[4] Ebbinghaus, H.-D., J. Flum, and W. Thomas, Mathematical Logic, Springer, New York, 1984.
Mathematical Reviews (MathSciNet): MR85a:03001
Zentralblatt MATH: 0556.03001
[5] Feferman, S., ``Transfinite recursive projections of axiomatic theories,'' The Journal of Symbolic Logic, vol. 27 (1962), pp. 259--316.
Mathematical Reviews (MathSciNet): MR30:3011
Zentralblatt MATH: 0117.25402
JSTOR: links.jstor.org
Digital Object Identifier: doi:10.2307/2964649
[6] Feferman, S., ``Introductory note to 1931c,'' Kurt Gödel: Collected works, vol. 1, edited by S. Feferman, Oxford University Press, London, 1986.
Mathematical Reviews (MathSciNet): MR831941
[7] Feferman, S. ``Reflecting on incompleteness,'' The Journal of Symbolic Logic, vol. 56 (1991), pp. 1--49.
Mathematical Reviews (MathSciNet): MR93b:03097
Zentralblatt MATH: 0746.03046
JSTOR: links.jstor.org
Digital Object Identifier: doi:10.2307/2274902
[8] Gentzen, G., ``New version of the consistency proof for elementary number theory,'' pp. 252--86 in The Collected Papers of Gerhard Gentzen, edited by M. E. Szabo, North-Holland, Amsterdam, 1969. (translated from the original: ``Neue Fassung der Wiederspruchfreiheitsbeweises für die Reine Zahlentheorie,'' Forschung zur Logic und zur Grundlegung der Exakten Wissenschaften, new series vol. 4 (1938), pp. 19--44).
Zentralblatt MATH: 0209.30001
[9] Graham, R. L., B. L. Rothschild, and J. H. Spencer, Ramsey Theory, 2d edition, Wiley Interscience Series in Discrete Mathematics and Optimization, New York, 1990.
Mathematical Reviews (MathSciNet): MR90m:05003
Zentralblatt MATH: 0705.05061
[10] Ignjatović, A., ``Hilbert's program and the $\omega$-rule,'' The Journal of Symbolic Logic, vol. 59 (1994), pp. 322--43.
Mathematical Reviews (MathSciNet): MR95f:03094
Zentralblatt MATH: 0820.03002
JSTOR: links.jstor.org
Digital Object Identifier: doi:10.2307/2275269
[11] Jones, J. P., and Y. Matijasevič, ``Proof of recursive unsolvability of Hilbert's tenth problem,'' American Mathematical Monthly, vol. 98 (1991), pp. 689--709.
Mathematical Reviews (MathSciNet): MR92i:03050
Zentralblatt MATH: 0746.03006
Digital Object Identifier: doi:10.2307/2324421
JSTOR: links.jstor.org
[12] Kotlarski, H., ``On the incompleteness theorems,'' The Journal of Symbolic Logic, vol. 59 (1994), pp. 1414--19.
Mathematical Reviews (MathSciNet): MR96c:03111
Zentralblatt MATH: 0816.03025
JSTOR: links.jstor.org
Digital Object Identifier: doi:10.2307/2275714
[13] Mendelson, E., Introduction to Mathematical Logic, Van Nostrand, Princeton, 1964.
Mathematical Reviews (MathSciNet): MR29:2158
Zentralblatt MATH: 0192.01901
[14] Paris, J., and L. Harrington, ``A mathematical incompleteness in Peano Arithmetic,'' pp. 1133--42 in Handbook of Mathematical Logic, edited by J. Barwise, North-Holland, Amsterdam, 1977.
Mathematical Reviews (MathSciNet): MR457132
[15] Smorinski, C., ``The incompleteness theorems,'' pp. 821--65 in Handbook of Mathematical Logic, edited by J. Barwise, North-Holland, Amsterdam, 1977.
[16] Smorinski, C., ``Some rapidly growing functions,'' Mathematical Intelligencer, vol. 2 (1980), pp. 149--154.
Mathematical Reviews (MathSciNet): MR595082
Zentralblatt MATH: 0453.03049
[17] Suppes, P., Axiomatic Set Theory, Dover, New York, 1972.
Mathematical Reviews (MathSciNet): MR50:1883
Zentralblatt MATH: 0269.02028
[18] Takeuti, G. Proof Theory, 2d edition, North-Holland, Amsterdam, 1987.
Mathematical Reviews (MathSciNet): MR89a:03115
Zentralblatt MATH: 0609.03019
[19] Turing, A., ``Systems of logic based on ordinals,'' Proceedings of the London Mathematical Society, series 2, vol. 45, (1939) pp. 161--228.
Zentralblatt MATH: 0021.09704
Notre Dame Journal of Formal Logic
