Location of Repository

Independence and conservativity results for intuitionistic set theory

By Ray-Ming Chen


There are two main parts to this thesis. The first part will deal with some independence results. In 1979, Lifschitz in [13] introduced a realizability interpretation\ud for Heyting's arithmetic, HA, that could differentiate between Church's thesis with uniqueness condition, CT0!, and the general form of Church's thesis, CT0. The objective here is to extend Lifschitz' realizability to intuitionistic Zermelo-Fraenkel set theory with two sorts, IZFN. In addition to separating Church's thesis with uniqueness condition from its general form in intuitionistic set theory, I also obtain several interesting\ud corollaries. The interpretation repudiates a weak form of countable choice, ACN2, asserting that every countable family of inhabited subsets of {0,1} has a choice function.\ud \ud The second part will be concerned with Constructive Zermelo-Fraenkel Set Theory and other intuitionistic set theories augmented by various principles, notably choice principles. It will be shown that the addition of these (choice) principles does not change the stock of provable arithmetical theorems.\ud \ud This type of conservativity result has its roots in a theorem of Goodman[9] who showed that Heyting arithmetic in all nite types augmented by the axiom of choice for all levels is conservative over HA. The technique I employ here to obtain such results for intuitionistic set theories, however, owes a lot to a paper by Beeson published in 1979. In [2] he showed how to construe Goodman's Theorem as the composition of two interpretations, namely relativized realizability and forcing. In this thesis, I adopt the same\ud approach and employ it to a plethora of intuitionistic set theories

Publisher: School of Mathematics (Leeds)
Year: 2010
OAI identifier: oai:etheses.whiterose.ac.uk:1439

Suggested articles



  1. (1980). Computability: An introduction to recursive function theory,
  2. (1979). Continuity in intuitionistic set theories,
  3. (1979). CT0 is stronger than CT0!,
  4. (1988). Dalen: Constructivism in Mathematics, Volumes I, II (North Holland,
  5. (1964). Feferman: the Number Systems: Foundations of Algebra and Analysis,
  6. (1970). Formal systems for some branches of intuitionistic analysis,
  7. (1970). Formal systems for some branches of intuitionistic analysis.
  8. (1980). Foundations of Constructive Mathematics,
  10. (2010). Independence and conservativity results for intuitionistic set theory,
  11. (1964). Introduction to Mathematical Logic,
  12. (1964). Introduction to Metamathematics (North-Holland,Amsterdam,
  13. (1964). Kleene: Introduction to Metamathematics,
  14. (1988). Mathematical Intuitionism-Introduction to Proof Theory
  15. (1988). Mathematical Intuitionism-Introduction to Proof Theory,
  16. (1986). McCarty:
  17. (1984). McCarty: Realizability and Recursive Mathematics,
  18. (2001). Notes on constructive set theory,
  19. (1982). Numbers, sets and axioms,
  20. (1945). On the interpretation of intuitionistic number theory.
  21. (1991). Oosten: Exercises in realizability,
  22. Oosten: Lifschitz' Realizability,
  23. (1990). Oosten: Lifschitz’s Realizability,
  24. (1984). Realizability and recursive mathematics, PhD thesis,
  25. (1986). Realizability and recursive set theory,
  26. (2006). Realizability for Constructive Zermelo-Fraenkel Set Theory,
  27. (2006). Realizability for constructive Zermelo-Fraenkel set theory.
  28. (1995). Realizability, Set Theory and Term Extraction, The Curry-Howard Isomorphism,
  29. (1947). Recursive Functions and Intuitionistic Number Theory,
  30. (1978). Relativized Realizability in Intuitionistic Arithmetic of All Finite Types,
  31. (1977). Set Theoretic Foundations for Constructive Analysis,
  32. (1973). Some applications of Kleene’s method for intuitionistic systems.
  33. (1973). Some properties of Intuitionistic Zermelo-Fraenkel set theory.
  34. (1998). The higher infinite in proof theory.
  35. (1995). The Mathematical Work of S.
  36. (1976). The theory of the G odel functionals,

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.