Sets in Types, Types in Sets

We present two mutual encodings, respectively of the Calculus of Inductive Constructions in Zermelo-Fraenkel set theory and the opposite way. More precisely, we actually construct two families of encodings, relating the number of universes in the type theory with the number of inaccessible cardinals in the set theory. The main result is that both hierarchies of logical formalisms interleave w.r.t. expressive power and thus are essentially equivalent. Both encodings are quite elementary: type theory is interpreted in set theory through a generalization of Coquand's simple proof-irrelevance interpretation. Set theory is encoded in type theory using a variant of Aczel's encoding; we have formally checked this last part using the Coq proof assistant.

[1]  P. Dybjer.  Inductive sets and families in Martin-Lo¨f's type theory and their set-theoretic semantics , 1991 .

[2]  J. Paris,et al.  The Type Theoretic Interpretation of Constructive Set Theory , 1978 .

[3]  Kenneth Kunen,et al.  Set Theory: An Introduction to Independence Proofs , 2010 .

[4]  Zhaohui Luo.  An extended calculus of constructions , 1990 .

[5]  Benjamin Werner,et al.  A Generic Normalisation Proof for Pure Type Systems , 1996, TYPES.

[6]  Thorsten Altenkirch,et al.  Constructions, inductive types and strong normalization , 1993, CST.

[7]  Thierry Coquand,et al.  Inductively defined types , 1988, Conference on Computer Logic.

[8]  T. Coquand,et al.  Metamathematical investigations of a calculus of constructions , 1989 .

[9]  J. Krivine,et al.  Théorie axiomatique des ensembles ... , 1972 .

[10]  Alonzo Church,et al.  A formulation of the simple theory of types , 1940, Journal of Symbolic Logic.

[11]  John C. Reynolds,et al.  Polymorphism is not Set-Theoretic , 1984, Semantics of Data Types.

[12]  Zhaohui Luo,et al.  ECC, an extended calculus of constructions , 1989, [1989] Proceedings. Fourth Annual Symposium on Logic in Computer Science.

[13]  Douglas J. Howe.  On computational open-endedness in Martin-Lof's type theory , 1991, [1991] Proceedings Sixth Annual IEEE Symposium on Logic in Computer Science.

[14]  Peter Hamburger,et al.  Set Theory: Introduction , 1999 .

[15]  P. Aczel.  The Type Theoretic Interpretation of Constructive Set Theory: Choice Principles , 1982 .

[16]  Per Martin-Löf,et al.  Intuitionistic type theory , 1984, Studies in proof theory.

[17]  Hugo Herbelin,et al.  The Coq proof assistant : reference manual, version 6.1 , 1997 .

[18]  Peter Aczel,et al.  The Type Theoretic Interpretation of Constructive Set Theory: Inductive Definitions , 1986 .

[19]  Paul Taylor,et al.  Intuitionistic sets and ordinals , 1996, Journal of Symbolic Logic.

[20]  Thierry Coquand,et al.  An Analysis of Girard's Paradox , 1986, LICS.