On the circularity of set-theoretic semantics for set theory

Journal title EPISTEMOLOGIA
Author/s Luca Bellotti
Publishing Year 2014 Issue 2014/1
Language English Pages 21 P. 58-78 File size 635 KB
DOI 10.3280/EPIS2014-001004
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The set-theoretic nature of the usual semantics of set theory raises a problem of circularity. A recourse to an intuitive semantics (possibly in terms of the iterative concept of set) is often deemed necessary, and a certain kind of realist philosophy of mathematics seems its best justification, taking for granted a well-determined reality of which settheoretic statements are true. I argue that, on the contrary, this form of realism leaves one in even deeper trouble. I try to understand the circularity of the set-theoretic semantics of set theory and the related crucial problem of quantification over the universe of sets in the light of a different, ‘Neo-Kantian’ perspective.

Keywords: Set theory, semantics, iterative concept of set, quantification over the universe, mathematical realism, Kreisel, Kant.

  1. Chihara C. (2004). A structural account of mathematics, Oxford, Oxford University Press.
  2. Dales H. G., Oliveri G. (1998). Truth and the foundations of mathematics: an introduction.
  3. In Dales H.G., Oliveri G. (eds.), Truth in mathematics, Oxford, Oxford University Press, pp. 1-37.
  4. Feferman S. (2004). Notes on operational set theory I (to appear).
  5. Gödel K. (1947/64). What is Cantor’s continuum problem?, American Mathematical Monthly, 54 (1947), pp. 515-525; revised version (1964) reprinted in Benacerraf and Putnam (1983), pp. 470-485; both versions reprinted in Gödel K. (1990), Collected works II, Oxford, Oxford University Press, pp. 176-188 and 254-270.
  6. Hallett M. (1984). Cantorian Set Theory and Limitation of Size, Oxford, Oxford University Press.
  7. Kant I. (1787). Kritik der reinen Vernunft, Riga, Johann Friedrich Hartknoch.
  8. Kreisel G. (1965). Mathematical Logic. In Saaty L. (ed.), Lectures in Modern Mathematics, New York, Wiley, pp. 95-195.
  9. Kreisel G. (1967). Informal Rigour and Completeness Proofs. In Lakatos I. (ed.), Problems in the philosophy of mathematics, Amsterdam, North Holland, pp. 138-186.
  10. Kreisel G. (1967a). Mathematical logic: what has it done for the philosophy of mathematics?. In Schoenman R. (ed.), Bertrand Russell, Philosopher of the Century, London, Allen & Unwin, pp. 201-272.
  11. Kreisel G., Krivine J. L. (1967). Eléments de Logique Mathématique, Paris, Dunod.
  12. Lawvere F. W., Rosebrugh R. (2003). Sets for mathematics, Cambridge, Cambridge University Press.
  13. Lear J. (1977). Sets and Semantics, Journal of Philosophy, 74, pp. 86-102.
  14. Maddy P. (1990). Realism in mathematics, Oxford, Oxford University Press.
  15. Bellotti L. (2010). A note on the circularity of set-theoretic semantics for set theory. In D’Agostino M. et al. (eds.), New Essays in Logic and Philosophy of Science, London, College Publications, pp. 207-215.
  16. Benacerraf P. (1973). Mathematical truth, Journal of Philosophy, 70, pp. 661-679; reprinted in Benacerraf and Putnam (1983), pp. 403-421.
  17. McGee V. (2000), ‘Everything’. In Sher G., Tieszen R. (eds.), Between logic and intuition, Cambridge, CUP, pp. 54-78.
  18. Moschovakis Y. N. (1980). Descriptive set theory, Amsterdam, North Holland.
  19. Oliveri G. (2007). A realist philosophy of mathematics, London, College Publications.
  20. Parsons C. (1974b). Informal axiomatization, formalization, and the concept of truth, Synthese, 27, pp. 27-47; reprinted in Parsons (1983), pp. 71-91. Parsons C. (1974c). Sets and classes, Noûs, 8, pp. 1-12; reprinted in Parsons (1983), pp. 209-220.
  21. Parsons C. (1974c). The liar paradox, Journal of Philosophical Logic, 3, pp. 381-412; reprinted in Parsons (1983), pp. 221-267. Parsons C. (1977). What is the iterative concept of set? In Butts R.E., Hintikka J. (eds.), Logic, foundations of mathematics and computability theory, Dordrecht, Reidel, pp. 335-367; reprinted in Benacerraf, Putnam (1983), pp. 503-529, and Parsons (1983), pp. 268-297.
  22. Parsons C. (1983). Mathematics in Philosophy, Ithaca, Cornell UP.
  23. Paseau A. (2001). Should the logic of set theory be intuitionistic?, Proceedings of the Aristotelian society, 101, pp. 369-378.
  24. Paseau A. (2003). The open-endedness of the set concept and the semantics of set theory, Synthese, 135, pp. 379-399.
  25. Scott D. (1974). Axiomatising set theory. In Jech T. (ed.), Axiomatic Set Theory II, Providence, AMS, pp. 207-214.
  26. Shoenfield J. R. (1967). Mathematical Logic, Reading, Addison-Wesley.
  27. Shoenfield J. R. (1977). Axioms of set theory. In Barwise J. (ed.), Handbook of Mathematical Logic, Amsterdam, North Holland, pp. 321-344.
  28. Wang H. (1974). From mathematics to philosophy, London, Routledge.
  29. Wang H. (1977). Large sets. In Butts R.E., Hintikka J. (eds.), Logic, foundations of mathematics and computability theory, Dordrecht, Reidel, pp. 309-333.
  30. Zermelo E. (1930). Über Grenzzahlen und Mengenbereiche, Fundamenta Mathematicae, 16, pp. 29-47.
  31. Benacerraf P., Putnam H. (eds.) (1983). Philosophy of mathematics, 2nd edition, Cambridge, Cambridge University Press.
  32. Boolos G. (1971). The iterative concept of set, Journal of Philosophy, 68, pp. 215-232; reprinted in Benacerraf, Putnam (1983), pp. 486-502.
  33. Boolos G. (1985). Nominalist Platonism, Philosophical Review, 94, pp. 327-344.
  34. Cassirer E. (1910). Substanzbegriff und Funktionsbegriff, Berlin, Bruno Cassirer; English translation, Chicago 1923.

Luca Bellotti, On the circularity of set-theoretic semantics for set theory in "EPISTEMOLOGIA" 1/2014, pp 58-78, DOI: 10.3280/EPIS2014-001004