How to Choose New Axioms for Set Theory?

  • Laura FontanellaEmail author
Part of the Synthese Library book series (SYLI, volume 407)


We address the problem of the choice of new axioms for set theory. After discussing some classical views about the notion of axiom in mathematics, we present the most currently debated candidates for a new axiomatisation of set theory, including Large Cardinal axioms, Forcing Axioms and Projective Determinacy and we illustrate some of the main arguments presented in favour or against such principles.



I would like to thank the anonymous reviewer for his careful reading and for his constructive comments. I am also indebted to Juliette Kennedy, Menachem Magidor, Neil Barton and Claudio Ternullo who gave me many useful suggestions that helped improve the quality of this paper.


  1. Aczel, P. (1988). Non-well-founded sets (CSLI Lecture Notes: Number 14). Stanford: CSLI Publications.Google Scholar
  2. Drake, F. (1974) Set theory. Amsterdam: North-Holland.Google Scholar
  3. Feferman, S., Friedman, H., Maddy, P., & Steel, J. (2000) Does mathematics need new axioms? Bulletin of Symbolic Logic, 6(4), 401–446.CrossRefGoogle Scholar
  4. Fraenkel, A., Bar-Hillel, Y., & Levy, A. (1973). Foundations of set theory (2nd ed.). Amsterdam: North-Holland.Google Scholar
  5. Forti, M., & Honsell, F. (1983). Set theory with free construction principles. Annali Scuola Normale Superiore di Pisa, Classe di Scienze, 10, 493–522.Google Scholar
  6. Gödel, K. (1947). What is Cantor continuum problem. The American Mathematical Montly, 54(9), 515–525.CrossRefGoogle Scholar
  7. Hamkins, J. D. (2014). A multiverse perspective on the axiom of constructibility. In: C.-T. Chong (Ed.), Infinity and truth (vol. 25, pp. 25–45). Hackensack: World Science Publication.CrossRefGoogle Scholar
  8. Kanamori, A., & Magidor, M. (1978). The evolution of large cardinal axioms in set theory. In: G. H. Muller & S. D. Scott (Eds.), Higher set theory (Lecture notes in Mathematics, vol. 669, pp. 99–275). Berlin: Springer.CrossRefGoogle Scholar
  9. Maddy, P. (1988). Believing the axioms, I. Journal of Symbolic Logic, 53, 481–511; II, ibid., 736–764.CrossRefGoogle Scholar
  10. Maddy, P. (2011). Defending the axioms: On the philosophical foundations of set theory. Oxford: Oxford University Press.CrossRefGoogle Scholar
  11. Magidor, M. (2012). Some set theories are more equal. Unpublished notes, available at Google Scholar
  12. Mayberry, J. P. (2000). The foundations of mathematics in the theory of sets. (Enciclopedia of mathematics and its applications, vol. 82). Cambridge: Cambridge University Press.Google Scholar
  13. Markov, A. A. (1962). On constructive mathematics. Trudy Matematicheskogo Instituta Imeni V. A. Steklova, 67(8–14). Translated in American Mathematical Society Translations: Series 2, 98, 1–9.Google Scholar
  14. McLarty, C. (2010). What does it take to prove Fermat’s last theorem? Grothendieck and the logic of number theory. The Bulletin of Symbolic Logic, 16(3), 359–377.CrossRefGoogle Scholar
  15. Moschovakis, Y. N. (1980). Descriptive set theory. Amsterdam: North-Holland.Google Scholar
  16. Nykos, P. J. (1980). A provisional solution to the normal moore space problem. Proceedings of the American Mathematical Society, 78(3), 429–435.CrossRefGoogle Scholar
  17. Reinhardt, W. N. (1974). Remarks on reflection principle large cardinals and elementary embeddings. Axiomatic set theory (Proceedings of symposia in pure mathematics, vol. XIII, pp. 189–205, Part II). Providence: American Mathematical Society.Google Scholar
  18. Scott, D. S. (1961). Measurable cardinals and constructible sets, Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, 7, 145–149.Google Scholar
  19. Shelah, S. (1974). Infinite Abelian groups, whitehead problem and some constructions. Israel Journal of Mathematics, 18(3), 243–256.CrossRefGoogle Scholar
  20. Solovay, R. M., Reinhardt, W. N., Kanamori, A. (1978). Strong axioms of infinity and elementary embeddings. Annals of Mathematical Logic, 13, 73–116.CrossRefGoogle Scholar
  21. Wang, H. (1974) The concept of set, in Benacerraf and Putnam [1983], pp. 530–570.Google Scholar

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Authors and Affiliations

  1. 1.Institut de Mathématiques de MarseilleUniversité Aix MarseilleCachanFrance

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