Finite Type Structures within Combinatory Algebras

Inside a combinatory algebra, there are 'internal' versions of the nite type structure over !, which form models of various systems of nite type arithmetic. This paper compares internal representations of the intensional and extensional functionals. If these classes coincide, the algebra is called ft-extensional. Some criteria for ft-extensionality are given and a number of well-known ca's are shown to be ft-extensional, regardless of the particular choice of representation for !. In particular, D A , P ! , T ! , H ! and certain D 1-models all share the property of ft-extensionality. It is also shown that ft-extensionality is by no means an intrinsic property of ca's, i.e. that there exists a very concrete class of ca's-the class of reeexive coherence spaces-no member of which has this property. This leads to a comparison of ft-extensionality with the well-studied notions of extensionality and weak extensionality. Ft-extensionality turns out to be completely independent .