Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-05-17T18:56:53.475Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  15 January 2014

Tatiana Arrigoni
Tatiana Arrigoni*
Affiliation:
Fondazione Bruno Kessler, Povo- Via Sommarive 18, 1-38123 Trento, ItalyE-mail: arrigoni@fbk.euandtatiana.arrigoni@istruzione.it
Get access Rights & Permissions [Opens in a new window]

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.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 17 , Issue 3 , September 2011 , pp. 337 - 360
Copyright
Copyright © Association for Symbolic Logic 2011

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

REFERENCES

[1] Arrigoni, Tatiana, What is meant by V? Reflections on the universe of all sets, Mentis Verlag, Paderborn, 2007.Google Scholar
[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. 43–64.Google Scholar
[3] Bell, John L, Boolean-valued models and independence proofs in set theory, Oxford University Press, Oxford, 1977.Google Scholar
[4] Benacerraf, Paul and Putnam, Hilary (editors), Philosophy of mathematics. Selected readings, second ed., Cambridge University Press, Cambridge, 1983.Google Scholar
[5] Boolos, George, The iterative conception of set, The Journal of Philosophy, vol. 68 (1971), pp. 215–231.Google Scholar
[6] Boolos, George, Iteration again, Philosophical Topics, vol. 42 (1989), pp. 5–21.CrossRefGoogle Scholar
[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. 257–268.Google Scholar
[8] Devlin, Keith, Constructibility, Springer, Berlin, 1984.CrossRefGoogle Scholar
[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.Google Scholar
[10] Friedman, Sy David, Cantor's set theory from a modern point of view, Jahresbericht der Deutschen Mathematiker-Vereinigung, vol. 104 (2002), pp. 165–170.Google Scholar
[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.125–153, reprinted in [4], pp. 447-469, and [9], pp. 119-141.Google Scholar
[12] Gödel, Kurt, What is Cantor's Continuum Problemi, American Mathematical Monthly, vol. 54 (1947), pp. 515–525, reprinted in [9], pp. 176-187.CrossRefGoogle Scholar
[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. 258–273, revised and expanded version of [12], reprinted in [4], pp. 470-485, and [9], pp. 254-269.Google Scholar
[14] Hallett, Michael, Cantorian set theory and limitation of size, Claredon Press, Oxford, 1984.Google Scholar
[15] Hauser, Kai, Objectivity over Objects. A case study in theory formation, Synthese, vol. 128 (2001), pp. 245–285.Google Scholar
[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. 93–117.Google Scholar
[17] Hauser, Kai, Is Choice self-evident?, American Philosophical Quarterly, vol. 42 (2005), pp. 237–261.Google Scholar
[18] Hauser, Kai, Gödel's program revisited. Part I. The turn to phenomenology, this Bulletin, vol. 12 (2006), pp. 529–590.Google Scholar
[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. 373–393.Google Scholar
[20] Jech, Thomas, Set theory, the third millenium, revised and expanded ed., Springer, Berlin, 2003.Google Scholar
[21] Jensen, Ronald, The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4 (1972), pp. 229–238.Google Scholar
[22] Jensen, Ronald, Inner models and large cardinals, this Bulletin, vol. 1 (1995), pp. 393–407.Google Scholar
[23] Kanamori, Akihiro, The mathematical development of set theory from Cantor to Cohen, this Bulletin, vol. 2 (1996), pp. 1–71.Google Scholar
[24] Kanamori, Akihiro, The higher infinite, second ed., Springer, Berlin, 2003.Google Scholar
[25] Kechris, Alexander S., Classical descriptive set theory, Springer, Berlin, 1995.Google Scholar
[26] Maddy, Penelope, Does V = L?, The Journal of Symbolic Logic, vol. 58 (1993), pp. 15–41.Google Scholar
[27] Maddy, Penelope, Naturalism in mathematics, Clarendon Press, Oxford, 1997.Google Scholar
[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. 134–152.Google Scholar
[29] Maddy, Penelope, Believing the axioms I, II, The Journal of Symbolic Logic, vol. 53 (1998), pp. 481-511 and 736–764.Google Scholar
[30] Maddy, Penelope, Mathematical existence, this Bulletin, vol. 11 (2005), pp. 351–376.Google Scholar
[31] Martin, Donald, Mathematical evidence, Truth in mathematics (Dales, H. G. and Olivieri, G., editors), Clarendon Press, Oxford, 1998, pp. 215–231.Google Scholar
[32] Mitchell, William J., Beginning inner model theory, Handbook of set theory (Foreman, Matthew and Kanamori, Akihiro, editors), Springer, Berlin, 2010, pp. 1449–1496.Google Scholar
[33] Moore, Gregory H., Zermelo's Axiom of Choice. Its origins and its development, Springer, Berlin, 1982.Google Scholar
[34] Moschovakis, Yiannis, Descriptive set theory, North Holland, Amsterdam, 1980.Google Scholar
[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. 335–367, reprinted in [4], pp. 503-529.Google Scholar
[36] Parsons, Charles, Platonism and mathematical intuition in Kurt Gödel's thought, this Bulletin, vol. 1 (1995), pp. 44–74.Google Scholar
[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. 74–92.Google Scholar
[38] Parsons, Charles, The structuralist view of mathematical objects, The philosophy of mathematics today (Hart, W. D., editor), Oxford University Press, Oxford, 1996, pp. 271–309.Google Scholar
[39] Parsons, Charles, Reason and intuition, Synthese, vol. 125 (2000), pp. 299–315.Google Scholar
[40] Schindler, Ralf D. and Zeman, Martin, Fine structure, Handbook of set theory (Foreman, Matthew and Kanamori, Akihiro, editors), Springer, Berlin, 2010, pp. 605–656.Google Scholar
[41] Shelah, Saharon, The future of set theory, Set theory of the reals. Israel Mathematical Conference Proceedings, 6 (Judah, H., editor), 1991, pp. 1–12.Google Scholar
[42] Shoenfield, Joseph, The axioms of set theory, Handbook of mathematical logic (Barwise, J., editor), North Holland, Amsterdam, 1977, pp. 321–44.Google Scholar
[43] Steel, John, Mathematics needs new axioms, this Bulletin, vol. 4 (2000), pp. 422–433.Google Scholar
[44] Steel, John, An outline of inner model theory, Handbook of set theory (Foreman, Matthew and Kanamori, Akihiro, editors), Springer, Berlin, 2010, pp. 1595–1684.CrossRefGoogle Scholar
[45] Heijenoort, Jan Van, From Frege to Gödel. A source book in mathematical logic, Harvard University Press, Cambridge MA, 1967.Google Scholar
[46] Wang, Hao, The concept of set, From mathematics to philosophy, Routledge and Kegan Paul, London, 1974, pp. 181–223.Google Scholar
[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.Google Scholar
[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].Google Scholar
[49] Zermelo, Ernst, Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre, Fundamenta Mathematicae, vol. 16 (1930), pp. 29–47.Google Scholar