An Axiomatic Theory for Partial Functions

We describe an axiomatic theory for the concept of one-place, partial function, where function is taken in its extensional sense. The theory is rather general, i.e., concepts like natural numbers and sets are definable, and topics as non-strictness and self application can be handled. It contains a model of the (extensional) lambda calculus, and commonly applied mechanisms (like currying, inductive definability) are possible. Furthermore, the theory is equi-consistent with and equally powerful as ZF-Set Theory. The theory (called Axiomatic Function Theory, AFT) is described in the language of classical first order predicate logic with equality and one non-logical predicate symbol for function application. By means of some notational conventions, we describe a method within this logic to handle undefinedness in a natural way.