Interpretation and Truth in Set Theory

  • Rodrigo A. FreireEmail author
Part of the Trends in Logic book series (TREN, volume 47)


The present paper is concerned with the presumed concrete or interpreted character of some axiom systems, notably axiom systems for usual set theory. A presentation of a concrete axiom system (set theory, for example) is accompanied with a conceptual component which, presumably, delimitates the subject matter of the system. In this paper, concrete axiom systems are understood in terms of a double-layer schema, containing the conceptual component as well as the deductive component, corresponding to the first layer and to the second layer, respectively. The conceptual component is identified with a criterion given by directive principles. Two lists of directive principles for set theory are given, and the two double-layer pictures of set theory that emerged from these lists are analyzed. Particular attention is paid to set-theoretic truth and the fixation of truth-values in each double-layer picture. The semantic commitments of both proposals are also compared, and distinguished from the usual notion of ontological commitment, which does not apply. The approach presented here to the problem of concrete axiom systems can be applied to other mathematical theories with interesting results. The case of elementary arithmetic is mentioned in passing.


  1. 1.
    Bernays, P. 1942. A System of Axiomatic Set Theory: Part III. Infinity and Enumerability. Analysis. The Journal Of Symbolic Logic.Google Scholar
  2. 2.
    Cantor, G. 1899. Letter to Dedekind, [translated in van Heijenoort 1967, 113–117].Google Scholar
  3. 3.
    Cantor, G. 1955. Contributions to the Founding of the Theory of Transfinite Numbers. Dover.Google Scholar
  4. 4.
    Cohen, P. 2008. Set Theory and the Continuum Hypothesis. Dover.Google Scholar
  5. 5.
    Ferreirós, J. 2012. On Arbitrary Sets and $ZFC$, Bulletin of Symbolic Logic.Google Scholar
  6. 6.
    Fraenkel, A., Y. Bar-Hillel, and A. Levy. 1973. Foundations of Set Theory. North-Holland.Google Scholar
  7. 7.
    Gödel, K. 1947, 1963. What is Cantor’s Continuum Problem [in Benacerraf and Putnam 1983, 470–485].Google Scholar
  8. 8.
    Gaifman, H. 2012. On Ontology and Realism in Mathematics. Review of Symbolic Logic.Google Scholar
  9. 9.
    Gödel, K. 1995. Collected Works, vol. III. Oxford.Google Scholar
  10. 10.
    McGee, V. 1997. How Can we Learn Mathematical Language, The Philosophical Review, vol. 106, no. 1, pp. 35–68.CrossRefGoogle Scholar
  11. 11.
    Moore, G. 2013. Zermelo’s Axiom of Choice. Dover.Google Scholar
  12. 12.
    Shoenfield, J. 2001. Mathematical Logic. ASL.Google Scholar
  13. 13.
    Tait, W. 2005. The Provenance of Pure Reason. Oxford.Google Scholar
  14. 14.
    Zermelo, E. 1908. Untersuschungen über due Grundlagen der Mengenlehre, I, Mathematische Annalen 65, 261–281 [translated in van Heijenoort 1967, 199–215].Google Scholar
  15. 15.
    Zermelo, E. 1930. Über Grenzzahlen und Mengenbereiche: Neue Untersuschungen über die Grundlagen der Mengenlehre. Fundamenta Mathematicae 16: 29–47.CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Philosophy DepartmentUniversity of BrasíliaBrasíliaBrazil

Personalised recommendations