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Does Mathematics Need New Axioms?

  • Solomon Feferman (a1), Harvey M. Friedman (a2), Penelope Maddy (a3) and John R. Steel (a4)

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.

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[1] Feferman, Solomon, In the light of logic, Oxford University Press, New York, 1998.
[2] Feferman, Solomon, Does mathematics need new axioms?, American Mathematical Monthly, vol. 106 (1999), pp. 99111.
[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. 254270, 1990.
[4] Maddy, Penelope, Naturalism in mathematics, Oxford University Press, Oxford, 1997.
[5] Maddy, Penelope, Some naturalistic reflections on set theoretic method, to appear in Topoi.
[6] Maddy, Penelope, Naturalism and the a priori, to appear in New essays on the a priori (Boghossian, P. and Peakcocke, C., editors).
[7] Maddy, Penelope, Naturalism: friends and foes, to appear in Philosophical Perspectives 15, Metaphysics 2001 (Tomberlin, J., editor).
[8] Moore, Gregory, Zermelo's axiom of choice, Springer-Verlag, New York, 1982.
[1] Gödel, Kurt F., What is Cantor's continuum problem?, American Mathematical Monthly, vol. 54 (1947), pp. 515525.
[2] Feferman, Solomon, Is Cantor necessary?, in In the light of logic, Oxford University Press, New York, 1998.
[3] Foreman, Matthew, Magidor, Menachem, and Shelah, Saharon, Martin's maximum, saturated ideals, and non-regular ultrafilters, Annals of Mathematics, vol. 127 (1988), pp. 147.
[4] Maddy, Penelope, Naturalism in mathematics, Oxford University Press, Oxford, 1997.
[5] Martin, Donald A., Mathematical evidence, Truth in mathematics (Dales, H. G. and Oliveri, G., editors), Clarendon Press, Oxford, 1998, pp. 215231.
[6] Martin, Donald A. and Steel, John R., A proof of projective determinacy, Journal of the American Mathematical Society, vol. 2 (1989), pp. 71125.
[7] Martin, Donald A. and Steel, John R., Iteration trees, Journal of the American Mathematical Society, vol. 7 (1994), pp. 173.
[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. 381394.
[1] Arnold, V., Atiyah, M., Lax, P., and Mazur, B., editors, Mathematics: Frontiers and perspectives, American Mathematical Society, 2000.
[2] Browder, F., editor, Mathematics into the twenty-first century, American Mathematical Society Centennial Publications, Volume II, 1992.
[3] Friedman, H., On the necessary use of abstract set theory, Advances in Mathematics, vol. 41 (09 1981), no. 3, pp. 209280.
[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. 229261.
[5] Friedman, H., 90: Two universes, Individual FOM Postings, http://www.math.psu.edu/simpson/fom/, 06 1, 2000.
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Bulletin of Symbolic Logic
  • ISSN: 1079-8986
  • EISSN: 1943-5894
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