The Type Free Lambda Calculus

Abstract

The λ-calculus represents a class of (partial) functions (λ-definable functions) on the integers that turns out to be the class of (partial) recursive functions. The equivalence between the Turing computable functions and the general recursive functions was originally proved via the λ-calculus: the general recursive functions are exactly the λ-definable functions as are the Turing computable functions. The equivalence between the λ-definable functions and the recursive functions was one of the arguments used by Church to defend his thesis proposing the identification of the intuitive class of effectively computable functions with the class of recursive functions; in fact one can give arguments for the Church's superthesis that states that for the functions involved this identification preserves the intensional character—that is, process of computation.