The lack of definable witnesses and provably recursive functions in intuitionistic set theories

Abstract Let ZFIR(ZFIC) be intuitionistic ZF set theory formulated with Replacement (resp. Collection). It is known that if ZFIR proves a sentence ∃xA(x), then there is a formula C(z) so that ZFIR proves ∃!zC(z) and ∃x(C(x) ∧ A(x)), the existence property. It is shown that ZFIC does not have the existence property, and thus ZFIR ⫋ ZFIC. This remains true even if one adds Dependent Choice and all true Σ1 sentence of ZF. It is known that ZF and ZFIc have the same provably recursive functions. It is also shown that this is not true for ZFIC and ZFIR.

[1]  Michael Beeson,et al.  Continuity in Intuitionistic Set Theories , 1979 .

[2]  A. Troelstra.  Metamathematical investigation of intuitionistic arithmetic and analysis , 1973 .

[3]  Harvey M. Friedman,et al.  Classically and intuitionistically provably recursive functions , 1978 .

[4]  Yehoshua Bar-Hillel,et al.  Mathematical Logic and Foundations of Set Theory Proceedings of an International Colloquium Under the Auspices of the Israel Academy of Sciences and Humanities, Jerusalem, 11-14 November 1968 , 1970 .

[5]  Andre Scedrov,et al.  Large sets in intuitionistic set theory , 1984, Ann. Pure Appl. Log..

[6]  H. Friedman.  Some applications of Kleene's methods for intuitionistic systems , 1973 .

[7]  Harvey M. Friedman,et al.  The consistency of classical set theory relative to a set theory with intu1tionistic logic , 1973, Journal of Symbolic Logic.

[8]  Saharon Shelah,et al.  On Power-like Models for Hyperinaccessible Cardinals , 1972, J. Symb. Log..

[9]  D. Scott,et al.  Sheaves and logic , 1979 .

[10]  Jr. Hartley Rogers.  Theory of Recursive Functions and Effective Computability , 1969 .

[11]  Michael P. Fourman,et al.  Sheaf models for set theory , 1980 .

[12]  Azriel Levy.  Definability in Axiomatic Set Theory II , 1970 .

[13]  Chen C. Chang,et al.  Model Theory: Third Edition (Dover Books On Mathematics) By C.C. Chang;H. Jerome Keisler;Mathematics , 1966 .

[14]  J. Myhill,et al.  Some properties of intuitionistic zermelo-frankel set theory , 1973 .

[15]  Saharon Shelah,et al.  On models with power-like orderings , 1972, Journal of Symbolic Logic.

[16]  R. J. Grayson,et al.  Heyting-valued models for intuitionistic set theory , 1979 .

[17]  Andre Scedrov,et al.  Set existence property for intuitionistic theories with dependent choice , 1983, Ann. Pure Appl. Log..

[18]  S. Feferman.  Some applications of the notions of forcing and generic sets , 1964 .