Formalism and intuition in computability

Abstract

The model of recursive functions in 1934–1936 was a deductive formal system. In 1936, Turing and in 1944, Post introduced more intuitive models of Turing machines and generational systems. When they both died prematurely in 1954, their informal approach was replaced again by the very formal Kleene T -predicate for another decade. By 1965, researchers could no longer read the papers. A second wave of intuition arose with the book by Rogers and Lachlan's revealing papers. A third wave of intuition has arisen from 1996 to the present with a return to the original meaning of computability in the sense of Turing and Gödel, and a return of ‘recursive’ to its original meaning of ‘inductive’ and the founding of Computability in Europe by Cooper and others.