Abstract

We consider the informal concept of "computability" or "effective calculability" andtwo of the formalisms commonly used to define it, "(Turing) computability" and "(general) recursiveness". We consider their origin, exact technical definition, concepts, history, general English meanings, how they became fixed in their present roles, how they were first and are now used, their impact on nonspecialists, how their use will affect the future content of the subject of computability theory, and its connection to other related areas. After a careful historical and conceptual analysis of computability and recursion we make several recommendations in section 7 about preserving the intensional di#erences between the concepts of "computability" and "recursion." Specifically we recommend that: the term "recursive" should no longer carry the additional meaning of "computable" or "decidable;" functions defined using Turing machines, register machines, or their variants should be calle...

Citations

914	On computable numbers, with an application to the ”Entscheidungsproblem - Turing - 1936
593	Introduction to Metamathematics - Kleene - 1971
308	Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. I. Monatshefte fr Mathematik und Physik - Gödel - 1931
214	An unsolvable problem of elementary number theory - Church - 1936
106	Recursively enumerable sets of positive integers and their decision problems - Post - 1944
92	Computability and Unsolvability - Davis - 1958
91	Mathematical Thought from Ancient to Modern Times - Kline - 1972
74	A note on the Entscheidungsproblem - Church - 1936
68	Formal Reduction of the General Combinatorial Decision Problem - Post - 1943
64	On notation for ordinal numbers - Kleene - 1938
63	1928], Grundzüge der theoretischen Logik - Hilbert, Ackermann
60	Recursive unsolvability of a problem of Thue - Post - 1947
57	General recursive functions of natural numbers, Mathematische Annalen 112 - Kleene - 1936
51	Recursive functionals and quantifiers of finite types - Kleene - 1959
43	Church’s Thesis and Principles for Mechanisms - Gandy - 1980
41	Lambda-definability and recursiveness - Kleene - 1936
38	On undecidable proposition of formal mathematical systems - Gödel - 1934
33	Recursive predicates and quantifiers - Kleene - 1943
32	Über das Unendliche. Mathematische Annalen, 95, 1926 161-190. Translated as ”On the infinite” in van Heijenoort, From Frege to Gödel: A source book in mathematical logic - Hilbert
32	Computing machinery and intelligence, Mind 59 - Turing - 1950
31	The upper semi-lattice of degrees of recursive unsolvability - Kleene, Post - 1954
28	Foundations of Recursion Theory - Platek - 1966
27	Arithmetices principia, nova methodo exposita - Peano
26	Absolutely unsolvable problems and relatively undecidable propositions: account of an anticipation - Post - 1965
24	Soare, Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets - I - 1987
22	Formal definitions in the theory of ordinal numbers - Church, Kleene - 1937
19	Die Grundlagen der Mathematik, Abhandlungen aus dem Mathematischen Seminar der Hamburgischen Universität 6 - Hilbert - 1928
17	Hierarchies of number-theoretic predicates - Kleene - 1955
15	Recursion on the Countable Functionals - Normann
14	On the forms of the predicates in the theory of constructive ordinals (second paper - Kleene - 1955
14	Arithmetical predicates and function quantifiers - Kleene - 1955
14	Origins of recursive function theory - Kleene - 1981
14	Begründung der elementaren Arithmetik durch die rekurrierende Denkweise ohne Anwendung scheinbarer Veränderlichen mit unendlichem Ausdehnungsbereich. Skrifter, Norske Videnskaps-Akademi i Oslo (= Videnskapsselskapet i Kristiania), Matematisk-Naturvidenska - Skolem - 1923
14	Turing, Systems of logic based on ordinals - M - 1939
13	Finite combinatory processes–formulation - Post - 1936
12	Über die Grundlagen der Logik und der Arithmetik - Hilbert - 1904
12	Turing-machine computable functionals of finite types - Kleene - 1962
11	The Confluence of Ideas - Gandy - 1994
10	The constructive second number class - Church - 1938
10	Über eine bisher noch nicht benütze Erweiterung des finiten Standpunktes - Gödel - 1958
9	Undecidable diophantine propositions - Gödel - 1995
9	Turing’s analysis of computability, and major applications of it - Kleene - 1988
9	Degrees of Unsolvability: Local and Global Theory - Lerman - 1983
9	Degrees of recursive unsolvability: preliminary report - Post - 1948
8	Turing: Computability and λ-definability - M - 1937
7	Mathematical logic and the origin of modern computers - Davis - 1988
7	Definability, automorphisms, and dynamic properties of computably enumerable sets - Harrington, Soare - 1996
7	The word problem in semi-groups with cancellation - Turing - 1950
6	Alan Turing: the enigma, (Burnett Books and - Hodges - 1983
6	Turing, Text of a lecture by Turing on June 24 - M - 1949