Skip to main content
×
×
Home

V = L and Intuitive Plausibility in set Theory. A Case Study

  • Tatiana Arrigoni (a1)
Abstract

What counts as an intuitively plausible set theoretic content (notion, axiom or theorem) has been a matter of much debate in contemporary philosophy of mathematics. In this paper I develop a critical appraisal of the issue. I analyze first R. B. Jensen's positions on the epistemic status of the axiom of constructibility. I then formulate and discuss a view of intuitiveness in set theory that assumes it to hinge basically on mathematical success. At the same time, I present accounts of set theoretic axioms and theorems formulated in non-strictly mathematical terms, e.g., by appealing to the iterative concept of set and/or to overall methodological principles, like unify and maximize, and investigate the relation of the latter to success in mathematics.

Copyright
References
Hide All
[1] Arrigoni, Tatiana, What is meant by V? Reflections on the universe of all sets, Mentis Verlag, Paderborn, 2007.
[2] Bagaria, Joan, Natural axioms of set theory and the continuum problem, Logic, Methodology and Philosophy of Science. Proceedings of the Twelfth International Congress (Hájek, P. et al., editors), King's College Publications, London, 2005, pp. 4364.
[3] Bell, John L, Boolean-valued models and independence proofs in set theory, Oxford University Press, Oxford, 1977.
[4] Benacerraf, Paul and Putnam, Hilary (editors), Philosophy of mathematics. Selected readings, second ed., Cambridge University Press, Cambridge, 1983.
[5] Boolos, George, The iterative conception of set, The Journal of Philosophy, vol. 68 (1971), pp. 215231.
[6] Boolos, George, Iteration again, Philosophical Topics, vol. 42 (1989), pp. 521.
[7] Boolos, George, Must we believe in set theory?, Between logic and intuition. Essays in honor of Charles Parsons (Sher, G. and Tieszen, R., editors), Cambridge University Press, Cambridge, 2000, pp. 257268.
[8] Devlin, Keith, Constructibility, Springer, Berlin, 1984.
[9] Feferman, S., Dawson, J., Kleene, S., Moore, G., and Van Heijenoort, J. (editors), Kurt Gödel. Collected works, Volume II., Oxford University Press, New York, 1990.
[10] Friedman, Sy David, Cantor's set theory from a modern point of view, Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 104 (2002), pp. 165170.
[11] Gödel, Kurt, Russell's mathematical logic, The philosophy of Bertrand Russell (Schillp, Paul A., editor), The Library of Living Philosophers, Northwestern University, Evanston, 1944, pp.125153, reprinted in [4], pp. 447-469, and [9], pp. 119-141.
[12] Gödel, Kurt, What is Cantor's Continuum Problemi, American Mathematical Monthly, vol. 54 (1947), pp. 515525, reprinted in [9], pp. 176-187.
[13] Gödel, Kurt, What is Cantor's Continuum Problem?, Philosophy of mathematics. Selected readings (Benacerraf, P. and Putnam, H., editors), Prentice-Hall, Englewood Cliffs, N.J., 1964, pp. 258273, revised and expanded version of [12], reprinted in [4], pp. 470-485, and [9], pp. 254-269.
[14] Hallett, Michael, Cantorian set theory and limitation of size, Claredon Press, Oxford, 1984.
[15] Hauser, Kai, Objectivity over Objects. A case study in theory formation, Synthese, vol. 128 (2001), pp. 245285.
[16] Hauser, Kai, Was sind und was sollen neue Axiome, One hundred year of Russell's paradox (Link, G., editor), De Gruyter, Berlin, 2004, pp. 93117.
[17] Hauser, Kai, Is Choice self-evident?, American Philosophical Quarterly, vol. 42 (2005), pp. 237261.
[18] Hauser, Kai, Gödel's program revisited. Part I. The turn to phenomenology, this Bulletin, vol. 12 (2006), pp. 529590.
[19] Jané, Ignacio, The iterative concept of set from a Cantorian perspective, Logic, Methodology and Philosophy of Science. Proceedings of the Twelfth International Congress (Hájek, P., Valdes-Villanueva, L., and Westerstahl, D., editors), King's College Publications, London, 2005, pp. 373393.
[20] Jech, Thomas, Set theory, the third millenium, revised and expanded ed., Springer, Berlin, 2003.
[21] Jensen, Ronald, The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4 (1972), pp. 229238.
[22] Jensen, Ronald, Inner models and large cardinals, this Bulletin, vol. 1 (1995), pp. 393407.
[23] Kanamori, Akihiro, The mathematical development of set theory from Cantor to Cohen, this Bulletin, vol. 2 (1996), pp. 171.
[24] Kanamori, Akihiro, The higher infinite, second ed., Springer, Berlin, 2003.
[25] Kechris, Alexander S., Classical descriptive set theory, Springer, Berlin, 1995.
[26] Maddy, Penelope, Does V = L?, The Journal of Symbolic Logic, vol. 58 (1993), pp. 1541.
[27] Maddy, Penelope, Naturalism in mathematics, Clarendon Press, Oxford, 1997.
[28] Maddy, Penelope, V = L and maximize, Logic Colloquium '95 (Makowski, J. A and Raave, E. V., editors), Springer-Verlag, Berlin, Heidelberg, New York, 1998, pp. 134152.
[29] Maddy, Penelope, Believing the axioms I, II, The Journal of Symbolic Logic, vol. 53 (1998), pp. 481-511 and 736764.
[30] Maddy, Penelope, Mathematical existence, this Bulletin, vol. 11 (2005), pp. 351376.
[31] Martin, Donald, Mathematical evidence, Truth in mathematics (Dales, H. G. and Olivieri, G., editors), Clarendon Press, Oxford, 1998, pp. 215231.
[32] Mitchell, William J., Beginning inner model theory, Handbook of set theory (Foreman, Matthew and Kanamori, Akihiro, editors), Springer, Berlin, 2010, pp. 14491496.
[33] Moore, Gregory H., Zermelo's Axiom of Choice. Its origins and its development, Springer, Berlin, 1982.
[34] Moschovakis, Yiannis, Descriptive set theory, North Holland, Amsterdam, 1980.
[35] Parsons, Charles, What is the maximum iterative concept of set?, Proceedings of the fifth Congress of Logic, Methodology and Philosophy of Science 1975. Part I: Logic, Foundation of Mathematics and Computability Theory (Butts, R. E. and Hintikka, J., editors), 1977, pp. 335367, reprinted in [4], pp. 503-529.
[36] Parsons, Charles, Platonism and mathematical intuition in Kurt Gödel's thought, this Bulletin, vol. 1 (1995), pp. 4474.
[37] Parsons, Charles, Structuralism and the concept of set, Morality and belief. Essays in honour of Ruth-Barcan Marcus (Sinnott-Armstrong, W., editor), Cambridge University Press, Cambridge, 1995, pp. 7492.
[38] Parsons, Charles, The structuralist view of mathematical objects, The philosophy of mathematics today (Hart, W. D., editor), Oxford University Press, Oxford, 1996, pp. 271309.
[39] Parsons, Charles, Reason and intuition, Synthese, vol. 125 (2000), pp. 299315.
[40] Schindler, Ralf D. and Zeman, Martin, Fine structure, Handbook of set theory (Foreman, Matthew and Kanamori, Akihiro, editors), Springer, Berlin, 2010, pp. 605656.
[41] Shelah, Saharon, The future of set theory, Set theory of the reals. Israel Mathematical Conference Proceedings, 6 (Judah, H., editor), 1991, pp. 112.
[42] Shoenfield, Joseph, The axioms of set theory, Handbook of mathematical logic (Barwise, J., editor), North Holland, Amsterdam, 1977, pp. 321–44.
[43] Steel, John, Mathematics needs new axioms, this Bulletin, vol. 4 (2000), pp. 422433.
[44] Steel, John, An outline of inner model theory, Handbook of set theory (Foreman, Matthew and Kanamori, Akihiro, editors), Springer, Berlin, 2010, pp. 15951684.
[45] Heijenoort, Jan Van, From Frege to Gödel. A source book in mathematical logic, Harvard University Press, Cambridge MA, 1967.
[46] Wang, Hao, The concept of set, From mathematics to philosophy, Routledge and Kegan Paul, London, 1974, pp. 181223.
[47] Woodin, Hugh, The Continuum Hypothesis, I-II, Notices of the American Mathematical Society, vol. 48 (2001), no. 7, pp. 567-76 and 681–90.
[48] Zermelo, Ernst, Untersuchungen über die Grundlagen der Mengenlehre, I, Mathematische Annalen, vol. 65 (1908), pp. 261–81, Page references are from the reprint in '45, pp. 199-215].
[49] Zermelo, Ernst, Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre, Fundamenta Mathematicae, vol. 16 (1930), pp. 2947.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Bulletin of Symbolic Logic
  • ISSN: 1079-8986
  • EISSN: 1943-5894
  • URL: /core/journals/bulletin-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed