Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-19T19:48:56.354Z Has data issue: false hasContentIssue false
  • English
  • Français

Does Mathematics Need New Axioms?

Published online by Cambridge University Press:  15 January 2014

Solomon Feferman ,
Harvey M. Friedman ,
Penelope Maddy  and
John R. Steel
Solomon Feferman
Affiliation:
Department of Mathematics, Stanford University Stanford, California 94305, USAE-mail:sf@csli.stanford.edu
Harvey M. Friedman
Affiliation:
Department of Mathematics, Ohio State UniversityColumbus, OHIO 43210, USAE-mail:friedman@math.ohio-state.eduURL: www.math.ohio-state.edu/~friedman/
Penelope Maddy
Affiliation:
Department of Logic and Philosophy of Science, University Of California, Irvine, California 92697-5100, USAE-mail:pjmaddy@uci.edu
John R. Steel
Affiliation:
Department of Mathematics, University of California, Berkeley, Berkeley, California 94720, E-mail:steel@math.berkeley.edu
Get access Rights & Permissions [Opens in a new window]

Extract

Does mathematics need new axioms? was the second of three plenary panel discussions held at the ASL annual meeting, ASL 2000, in Urbana-Champaign, in June, 2000. Each panelist in turn presented brief opening remarks, followed by a second round for responding to what the others had said; the session concluded with a lively discussion from the floor. The four articles collected here represent reworked and expanded versions of the first two parts of those proceedings, presented in the same order as the speakers appeared at the original panel discussion: Solomon Feferman (pp. 401–413), Penelope Maddy (pp. 413–422), John Steel (pp. 422–433), and Harvey Friedman (pp. 434–446). The work of each author is printed separately, with separate references, but the portions consisting of comments on and replies to others are clearly marked.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 6 , Issue 4 , December 2000 , pp. 401 - 446
Copyright
Copyright © Association for Symbolic Logic 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCE

[1] Aczel, P. and Richter, W., Inductive definitions and analogues of large cardinals, in Conference in mathematical logic — London '70 (Hodges, W., editor), Lecture Notes in Mathematics, vol. 255, Springer-Verlag, Berlin, 1972, pp. 1–10.Google Scholar
[2] Feferman, S., Göodel's program for new axioms: Why, where, how and what?, in Göodel '96, (Hájek, P., editor), Lecture Notes in Logic, vol. 6, 1996, pp. 3–22.Google Scholar
[3] Feferman, S., In the light of logic, Oxford University Press, New York, 1998.CrossRefGoogle Scholar
[4] Feferman, S., Does mathematics need new axioms?, American Mathematical Monthly, vol. 106 (1999), pp. 99–111.Google Scholar
[5] Friedman, H., Finite functions and the necessary use of large cardinals, Annals of Mathematics, vol. 148 (1998), pp. 803–893.CrossRefGoogle Scholar
[6] Gödel, K., Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte für Mathematik und Physik, vol. 38 (1931), pp. 173–198. Reprinted, with English translation, in Collected Works,, Vol. I., Publications 1929–1936 (S. Feferman, et al., editors), Oxford University Press, New York, 1986, pp. 144–195.Google Scholar
[7] Gödel, K., What is Cantor's continuum problem?, American Mathematical Monthly, vol. 54 (1947), pp. 515–525; errata 55, 151. Reprinted in Collected Works, Vol. II., Publications 1938–1974 (S. Feferman, et al., editors), Oxford University Press, New York, 1990, pp. 176–187. (1964 revised and expanded version, ibid., pp. 254–270.)Google Scholar
[8] Gödel, K., Collected Works, Vol. III., Unpublished essays and lectures (S. Feferman, et al., editors), Oxford University Press, New York, 1995.Google Scholar
[9] Griffor, E. and Rathjen, M., The strength of some Martin-Löf type theories, Archive for Mathematical Logic, vol. 33 (1994), pp. 347–385.CrossRefGoogle Scholar
[10] Jäger, G. and Studer, T., Extending the system T0 of explicit mathematics: the limit and Mahlo axioms, to appear.Google Scholar
[11] Kanamori, A., The higher infinite, Springer-Verlag, Berlin, 1994.Google Scholar
[12] Maddy, P., Realism in mathematics, Clarendon Press, Oxford, 1990.Google Scholar
[13] Maddy, P., Naturalism in mathematics, Clarendon Press, Oxford, 1997.Google Scholar
[14] Martin, D. A., Hilbert's first problem: The continuum hypothesis, in Mathematical developments arising from Hilbert problems (Browder, F., editor), Proceedings of Symposia in Pure Mathematics, vol. 28, American Mathematical Society, Providence, 1976, pp. 81–92.Google Scholar
[15] Martin, D. A. and Steel, J., A proof of projective determinacy, Journal of the American Mathematical Society, vol. 2 (1989), pp. 71–125.CrossRefGoogle Scholar
[16] Paris, J. and Harrington, L., A mathematical incompleteness in Peano arithmetic, in Handbook of mathematical logic (Barwise, J., editor), North-Holland, Amsterdam, 1977, pp. 1133–1142.Google Scholar
[17] Pohlers, W., Subsystems of set theory and second order number theory, in Handbook of proof theory (Buss, S. R., editor), Elsevier, Amsterdam, 1998, pp. 209–335.Google Scholar
[18] Rathjen, M., Recent advances in ordinal analysis: -CA and related systems, Bulletin of Symbolic Logic, vol. 1 (1995), pp. 468–485.Google Scholar
[19] Simpson, S., Subsystems of second order arithmetic, Springer-Verlag, Berlin, 1998.Google Scholar
[20] Ye, F., Strict constructivism and the philosophy of mathematics, Ph. D. Dissertation , Princeton University, 1999.Google Scholar
[1] Feferman, Solomon, In the light of logic, Oxford University Press, New York, 1998.Google Scholar
[2] Feferman, Solomon, Does mathematics need new axioms?, American Mathematical Monthly, vol. 106 (1999), pp. 99–111.CrossRefGoogle Scholar
[3] Gödel, Kurt, What is Cantor's continuum problem?, reprinted in his Collected works, volume II (Feferman, S. et al., editors), Oxford University Press, New York, pp. 254–270, 1990.Google Scholar
[4] Maddy, Penelope, Naturalism in mathematics, Oxford University Press, Oxford, 1997.Google Scholar
[5] Maddy, Penelope, Some naturalistic reflections on set theoretic method, to appear in Topoi.Google Scholar
[6] Maddy, Penelope, Naturalism and the a priori, to appear in New essays on the a priori (Boghossian, P. and Peakcocke, C., editors).Google Scholar
[7] Maddy, Penelope, Naturalism: friends and foes, to appear in Philosophical Perspectives 15, Metaphysics 2001 (Tomberlin, J., editor).Google Scholar
[8] Moore, Gregory, Zermelo's axiom of choice, Springer-Verlag, New York, 1982.Google Scholar
[1] Gödel, Kurt F., What is Cantor's continuum problem?, American Mathematical Monthly, vol. 54 (1947), pp. 515–525.Google Scholar
[2] Feferman, Solomon, Is Cantor necessary?, in In the light of logic, Oxford University Press, New York, 1998.Google Scholar
[3] Foreman, Matthew, Magidor, Menachem, and Shelah, Saharon, Martin's maximum, saturated ideals, and non-regular ultrafilters, Annals of Mathematics, vol. 127 (1988), pp. 1–47.Google Scholar
[4] Maddy, Penelope, Naturalism in mathematics, Oxford University Press, Oxford, 1997.Google Scholar
[5] Martin, Donald A., Mathematical evidence, Truth in mathematics (Dales, H. G. and Oliveri, G., editors), Clarendon Press, Oxford, 1998, pp. 215–231.Google Scholar
[6] Martin, Donald A. and Steel, John R., A proof of projective determinacy, Journal of the American Mathematical Society, vol. 2 (1989), pp. 71–125.Google Scholar
[7] Martin, Donald A. and Steel, John R., Iteration trees, Journal of the American Mathematical Society, vol. 7 (1994), pp. 1–73.Google Scholar
[8] Shelah, Saharon and Woodin, W. Hugh, Large cardinals imply that every reasonably definable set of reals is Lebesgue measurable, Israel Journal of Mathematics, vol. . 70 (1990), pp. 381–394.Google Scholar
[1] Arnold, V., Atiyah, M., Lax, P., and Mazur, B., editors, Mathematics: Frontiers and perspectives, American Mathematical Society, 2000.Google Scholar
[2] Browder, F., editor, Mathematics into the twenty-first century, American Mathematical Society Centennial Publications, Volume II, 1992.Google Scholar
[3] Friedman, H., On the necessary use of abstract set theory, Advances in Mathematics, vol. 41 (09 1981), no. 3, pp. 209–280.Google Scholar
[4] Friedman, H., Robertson, N., and Seymour, P., The metamathematics of the graph minor theorem, in Logic and combinatorics (Simpson, S., editor), American Mathematical Society Contemporary Mathematics Series, vol. 65, 1987, pp. 229–261.Google Scholar
[5] Friedman, H., 90: Two universes, Individual FOM Postings, http://www.math.psu.edu/simpson/fom/, 06 1, 2000.Google Scholar