Abstract. In §1 we give a short overview for a general audience of Godel, Church, Turing, and the discovery of computability in the 1930s. In the later sections we mention a series of our previous papers where a more detailed analysis of computability, Turing's work, and extensive lists of references can be found. The sections from §2—§9 challenge the conventional wisdom and traditional ideas found in many books and papers on computability theory. They are based on a half century of my study of the subject beginning with Church at Princeton in the 1960s, and on a careful rethinking of these traditional ideas.

The references in all my papers and books are given in the format, author [year], as in Turing [1936], in order that the references are easily identified without consulting the bibliography and are uniform over all papers. A complete bibliography of historical articles from all my books and papers on computabilityis given on the page as explained in §10.

§1. A very brief overview of computability.

1.1. Hilbert's programs. Around 1880 Georg Cantor, a German mathematician, invented naive set theory. A small fraction of this is sometimes taught to elementary school children. It was soon discovered that this naive set theory was inconsistent because it allowed unbounded set formation, such as the set of all sets. David Hilbert, the world's foremost mathematician from 1900 to 1930, defended Cantor's set theory but suggested a formal axiomatic approach to eliminate the inconsistencies. He proposed two programs.