Towards an arithmetic for partial computable functionals

The thesis concerns itself with nonflat Scott information systems as an appropriate denotational semantics for the proposed theory TCF+, a constructive theory of higher-type partial computable functionals and approximations. We prove a definability theorem for type systems with at most unary constructors via atomic-coherent information systems, and give a simple proof for the density property for arbitrary finitary type systems using coherent information systems. We introduce the notions of token matrices and eigen-neighborhoods, and use them to locate normal forms of neighborhoods, as well as to demonstrate that even nonatomic information systems feature implicit atomicity. We then establish connections between coherent information systems and various pointfree structures. Finally, we introduce a fragment of TCF+ and show that extensionality can be eliminated.