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CIRCULARITY IN SOUNDNESS AND COMPLETENESS

Published online by Cambridge University Press:  13 May 2014

RICHARD KAYE
RICHARD KAYE*
Affiliation:
SCHOOL OF MATHEMATICS UNIVERSITY OF BIRMINGHAM BIRMINGHAM B15 2TT, UKE-mail:r.w.Kaye@bham.ac.uk
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Abstract

We raise an issue of circularity in the argument for the completeness of first-order logic. An analysis of the problem sheds light on the development of mathematics, and suggests other possible directions for foundational research.

Type
Articles
Information
Bulletin of Symbolic Logic , Volume 20 , Issue 1 , March 2014 , pp. 24 - 38
Copyright
Copyright © The Association for Symbolic Logic 2014 

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References

REFERENCES

Bernays, Paul, Beiträge zur axiomatischen Behandlung des Logik-Kalküls. Habilitattionsschrift, Universität Göttingen, 1918.Google Scholar
Carroll, Lewis, What the tortoise said to Achilles. Mind, vol. 104 (1995), no. 416, pp. 691–693.CrossRefGoogle Scholar
Michael, A. E.Dummett, The justification of deduction, Truth and Other Enigmas, British Academy Lecture, 1973, Duckworth, 1978, pp. 290–318.Google Scholar
Feferman, Solomon, Kreisel’s “unwinding” program, Kreiseliana (Odifreddi, Piergiorgio, editor), A K Peters, Wellesley, MA, 1996, pp. 247–273.Google Scholar
Field, Hartry, Saving truth from paradox. Oxford University Press, Oxford, 2008.Google Scholar
Friedman, Harvey, Countable models of set theories, Cambridge Summer School in Mathematical Logic, Cambridge, 1971, Lecture Notes in Mathematics, vol. 337. Springer, Berlin, 1973, pp. 539–573.CrossRefGoogle Scholar
Gödel, Kurt, Über die Vollständigkeit der Axiome des logischen Funktionenkalküs. Monatshefte für Mathematik und Physik, vol. 37 (1930), pp. 349–360.CrossRefGoogle Scholar
Gödel, Kurt, The completeness of the axioms of the functional calculus of logic, From Frege to Gödel. A source book in mathematical logic, 1879–1931 (Heijenoort, Jean van, editor), Harvard University Press, Cambridge, MA, 1967, pp. 582–591. Translation of Gödel [7].Google Scholar
Hilbert, David, Prinzipien der mathematik, Lecture notes by Paul Bernays. Bibliotek, Mathematisches Institut, Universität Göttingen, 1917-8.Google Scholar
Kaye, Richard, Kossak, Roman, and Wong, Tin Lok, Adding standardness to nonstandard arithmetic, New studies in weak arithmetics (Cégielski, P., Cornaros, Ch, and Dimitracopoulos, C., editors), CSLI Lecture Notes number 211, CSLI Publications, Stanford, 2014, pp. 179–197.Google Scholar
Kreisel, Georg, Informal rigour and completeness proofs, Problems in the Philosophy of Mathematics (Lakatos, Imre, editor), Problems in the Philosophy of Mathematics. North-Holland, 1967.Google Scholar
Lindström, Per, On extensions of elementary logic. Theoria, vol. 35 (1969), pp. 1–11.CrossRefGoogle Scholar
Paris, Jeff and Harrington, Leo, A mathematical incompleteness in Peano arithmetic, Handbook of Mathematical Logic, North-Holland Publishing Co., Amsterdam, 1977, pp. 1133–1142. Edited by Jon Barwise, With the cooperation of Keisler, H. J., Kunen, K., Moschovakis, Y. N. and Troelstra, A. S., Studies in Logic and the Foundations of Mathematics, Vol. 90.Google Scholar
Popper, Karl R., The logic of scientific discovery, Hutchinson and Co., Ltd., London, 1959.Google Scholar
Post, Emil, Introduction to a general theory of elementary propositions. American Journal of Mathematics, vol. 43 (1921), pp. 163–85.Google Scholar
Scott, Dana, Algebras of sets binumerable in complete extensions of arithmetic, Proceedings of Symposia in Pure Mathematics, vol. V, American Mathematical Society, Providence, RI, pp. 117–121, 1962.Google Scholar
Shelah, Saharon, Infinite abelian groups, Whitehead problem and some constructions. Israel Journal of Mathematics, vol. 18 (1974), pp. 243–256.Google Scholar
Simpson, Stephen G., Subsystems of second order arithmetic, second ed., Perspectives in Logic. Cambridge University Press, Cambridge, 2009.Google Scholar
Stolboushkin, Alexei P., Towards recursive model theory, Logic Colloquium ’95 (Haifa), vol. 11 Lecture Notes Logic, Springer, Berlin, 1998, pp. 325–338.CrossRefGoogle Scholar
Tennenbaum, Stanley, Non-archimedian models for arithmetic. Notices of the American Mathematical Society, vol. 270 (1959), p. 270.Google Scholar
Zach, Richard, Completeness before Post: Bernays, Hilbert, and the development of propositional logic, this Journal, vol. 5 (1999), no. 3, pp. 331–366.Google Scholar