Constructive Zermelo-Fraenkel Set Theory, Power Set, and the Calculus of Constructions

  • Michael Rathjen
Part of the Logic, Epistemology, and the Unity of Science book series (LEUS, volume 27)


Full intuitionistic Zermelo-Fraenkel set theory, IZF, is obtained from constructive Zermelo-Fraenkel set theory, CZF, by adding the full separation axiom scheme and the power set axiom. The strength of CZFplus full separation is the same as that of second order arithmetic, using a straightforward realizability interpretation in classical second order arithmetic and the fact that second order Heyting arithmetic is already embedded in CZFplus full separation. This paper is concerned with the strength of CZFaugmented by the power set axiom, \({\mathbf{CZF}}_{\mathcal{P}}\). It will be shown that it is of the same strength as Power Kripke–Platek set theory, \(\mathbf{KP}(\mathcal{P})\), as well as a certain system of type theory, \({\mathbf{MLV}}_{\mathbf{P}}\), which is a calculus of constructions with one universe. The reduction of \({\mathbf{CZF}}_{\mathcal{P}}\)to \(\mathbf{KP}(\mathcal{P})\)uses a realizability interpretation wherein a realizer for an existential statement provides a set of witnesses for the existential quantifier rather than a single witness. The reduction of \(\mathbf{KP}(\mathcal{P})\)to \({\mathbf{CZF}}_{\mathcal{P}}\)employs techniques from ordinal analysis which, when combined with a special double negation interpretation that respects extensionality, also show that \(\mathbf{KP}(\mathcal{P})\)can be reduced to CZFwith the negative power set axiom. As CZFaugmented by the latter axiom can be interpreted in \({\mathbf{MLV}}_{\mathbf{P}}\)and this type theory has a types-as-classes interpretation in \({\mathbf{CZF}}_{\mathcal{P}}\), the circle will be completed.


Type Theory Bound Separation Realizability Interpretation Existential Quantifier Order Arithmetic 
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  1. Aczel, P. 1978. The type theoretic interpretation of constructive set theory. In Logic Colloquium ’77, ed. A. MacIntyre, L. Pacholski, and J. Paris. Amsterdam: North–Holland.Google Scholar
  2. Aczel, P. 1982. The type theoretic interpretation of constructive set theory: Choice prinicples. In The L.E.J. brouwer centenary symposium, ed. A.S. Troelstra and D. van Dalen. Amsterdam: North–Holland.Google Scholar
  3. Aczel, P. 1986. The type theoretic interpretation of constructive set theory: Inductive definitions. In Logic, methodology and philosophy science VII, ed. R.B. Marcus et al., 17–49. Amsterdam: North–Holland.Google Scholar
  4. Aczel, P. 2000. On relating type theories and set theories. In Types ’98, Lecture notes in computer science, vol. 1257, ed. T. Altenkirch, W. Naraschewski, and B. Reus. Berlin: Springer.Google Scholar
  5. Aczel, P., and M. Rathjen. 2001. Em notes on constructive set theory, Technical report 40, Institut Mittag-Leffler. Stockholm: The Royal Swedish Academy of Sciences.
  6. Aczel, P., and M. Rathjen. 2010. Constructive set theory, book draft.Google Scholar
  7. Barwise, J. 1975. Admissible sets and structures. Berlin/Heidelberg/New York: Springer.Google Scholar
  8. Beeson, M. 1985. Foundations of constructive mathematics. Berlin: Springer.CrossRefGoogle Scholar
  9. Chen, R.-M., and M. Rathjen. 2010. Lifschitz realizability for intuitionistic Zermelo-Fraenkel set theory, submitted.Google Scholar
  10. Coquand, T. 1990. Metamathematical investigations of a calculus of constructions. In Logic and Computer science, ed. P. Oddifreddi, 91–122. London: Academic.Google Scholar
  11. Friedman, H. 1973a. Some applications of Kleene’s method for intuitionistic systems. In Cambridge summer school in mathematical logic, Lectures notes in mathematics, vol. 337, ed. A. Mathias and H. Rogers, 113–170. Berlin: Springer.Google Scholar
  12. Friedman, H. 1973b. Countable models of set theories. In Cambridge summer school in mathematical logic, Lectures Notes in Mathematics, vol. 337, ed. A. Mathias and H. Rogers, 539–573. Berlin: Springer.Google Scholar
  13. Friedman, H. 1973c. The consistency of classical set theory relative to a set theory with intuitionistic logic. Journal of Symbolic Logic38: 315–319.CrossRefGoogle Scholar
  14. Feferman, S. 1979. Constructive theories of functions and classes. In Logic colloquium ’78, ed. M. Boffa, D. van Dalen, and K. McAloon, 1–52. Amsterdam: North-Holland.Google Scholar
  15. Gambino, N. 1999. Types and sets: A study on the jump to full impredicativity, Laurea dissertation, Department of Pure and Applied Mathematics, University of Padua.Google Scholar
  16. Lifschitz, V. 1979. CT0 is stronger than CT0! Proceedings of the American Mathematical Society73: 101–106.Google Scholar
  17. Lubarsky, R.S. 2006. CZF and second order arithmetic. Annals of Pure and Applied Logic141: 29–34.CrossRefGoogle Scholar
  18. Mac Lane, S. 1992. Form and function. Berlin: Springer.Google Scholar
  19. Martin-Löf, P. 1984. Intuitionistic type theory. Naples: Bibliopolis.Google Scholar
  20. Mathias, A.R.D. 2001. The strength of Mac Lane set theory. Annals of Pure and Applied Logic110: 107–234.CrossRefGoogle Scholar
  21. Moschovakis, Y.N. 1976. Recursion in the universe of sets, mimeographed note.Google Scholar
  22. Moss, L. 1995. Power set recursion. Annals of Pure and Applied Logic71: 247–306.CrossRefGoogle Scholar
  23. Myhill, J. 1975. Constructive set theory. Journal of Symbolic Logic40: 347–382.CrossRefGoogle Scholar
  24. Normann, D. 1978. Set recursion. In Generalized recursion theory II, 303–320. Amsterdam: North-Holland.Google Scholar
  25. Palmgren, E. 1993. Type-theoretic interpretations of iterated, strictly positive inductive definitions. Archive for Mathematical Logic32: 75–99.CrossRefGoogle Scholar
  26. Pozsgay, L. 1971. Liberal intuitionism as a basis for set theory, in Axiomatic set theory. Proceedings Symposium Pure Mathematics12(1): 321–330.Google Scholar
  27. Pozsgay, L. 1972. Semi-intuitionistic set theory. Notre Dame Journal of Formal Logic13: 546–550.CrossRefGoogle Scholar
  28. Rathjen, M. 1994. The strength of some Martin-Löf type theories. Archive for Mathematical Logic33: 347–385.CrossRefGoogle Scholar
  29. Rathjen, M. 2005. Replacement versus collection in constructive Zermelo-Fraenkel set theory. Annals of Pure and Applied Logic136: 156–174.CrossRefGoogle Scholar
  30. Rathjen, M. 2006a. Choice principles in constructive and classical set theories. In Logic colloquium 2002, Lecture notes in logic, vol. 27, ed. Z. Chatzidakis, P. Koepke, and W. Pohlers, 299–326. Wellesley: A.K. Peters.Google Scholar
  31. Rathjen, M. 2006b. Realizability for constructive Zermelo-Fraenkel set theory. In Logic Colloquium 2003, Lecture notes in logic, vol. 24, ed. J. Väänänen and V. Stoltenberg-Hansen, 282–314. Wellesley: A.K. Peters.Google Scholar
  32. Rathjen, M. 2006c. The formulae-as-classes interpretation of constructive set theory. In Proof technology and computation, Proceedings of the international summer school marktoberdorf 2003, ed. H. Schwichtenberg and K. Spies, 279–322. Amsterdam: IOS Press.Google Scholar
  33. Rathjen, M., and S. Tupailo. 2006. Characterizing the interpretation of set theory in Martin-Löf type theory. Annals of Pure and Applied Logic141: 442–471.CrossRefGoogle Scholar
  34. Rathjen, M. 2011. An ordinal analysis of Power Kripke–Platek set theory, Centre de Recerca Matemàtica Barcelona preprint series.Google Scholar
  35. Rathjen, M. The weak existence property for intuitionistic set theories.Google Scholar
  36. Rathjen, M., and A. Weiermann. 1993. Proof-theoretic investigations on Kruskal’s theorem. Annals of Pure and Applied Logic60: 49–88.CrossRefGoogle Scholar
  37. Sacks, G.E. 1990. Higher recursion theory. Berlin: Springer.Google Scholar
  38. Tharp, L. 1971. A quasi-intuitionistic set theory. Journal of Symbolic Logic36: 456–460.CrossRefGoogle Scholar
  39. Thiele, E.J. 1968. Über endlich axiomatisierbare Teilsysyteme der Zermelo-Fraenkel’schen Mengenlehre. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik14: 39–58.CrossRefGoogle Scholar
  40. Troelstra, A.S. 1998. Realizability. In Handbook of proof theory, ed. S.R. Buss, 407–473. Amsterdam: Elsevier.CrossRefGoogle Scholar
  41. Troelstra, A.S., and D. van Dalen. 1988. Constructivism in mathematics, volumes I, II. Amsterdam: North Holland.Google Scholar
  42. van Oosten, J. 1990. Lifschitz’s realizability. The Journal of Symbolic Logic55: 805–821.CrossRefGoogle Scholar
  43. Wolf, R.S. 1974. Formally intuitionistic set theories with bounded predicates decidable. Ph.D. thesis, Stanford University.Google Scholar

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© Springer Science+Business Media Dordrecht. 2012

Authors and Affiliations

  1. 1.Department of Pure MathematicsUniversity of LeedsLeedsUK

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