## Church’s Thesis

### BibTeX

@MISC{Bezhanishvili_church’sthesis,

author = {Guram Bezhanishvili},

title = {Church’s Thesis},

year = {}

}

### OpenURL

### Abstract

In this project we will learn about both primitive recursive and general recursive functions. We will also learn about Turing computable functions, and will discuss why the class of general recursive functions coincides with the class of Turing computable functions. We will introduce the effectively calculable functions, and the ideas behind Alonzo Church’s (1903–1995) proposal to identify the

### Citations

1236 |
On computable numbers, with an application to the Entscheidungsproblem
- Turing
- 1936
(Show Context)
Citation Context ...he term “primitive recursive” was introduced by Kleene in [7]. 2 The Normal Form Theorem appeared in [7] and considerably simplified the notion of general recursive functions. 2Turing’s famous paper =-=[15]-=- appeared in 1936 (a correction to it was published in 1937). Turing introduced what we now call Turing machines, and defined a function to be computable if it can be computed on a Turing machine. His... |

850 |
Jr.; Theory of Recursive Functions and Effective Computability
- Rogers
- 1967
(Show Context)
Citation Context ...ed about recursive functions via a treatment that emphasized partial functions from the outset to realize just how important Kleene's contribution was. Thus Rogers' excellent and influential treatise =-=[14]-=-, p. 12, contains an historical account which gives the impression that the subject had been formulated in terms of partial functions from the beginning. To summarize, in the mid-thirties there were s... |

728 |
Introduction to Metamathematics
- KLEENE
- 1971
(Show Context)
Citation Context ...This footnote occurs in the original source. p q f p f o 94.(c) In your own words explain Kleene’s definition of a Turing machine by reading carefully the following excerpt from section 67 of Kleene =-=[11]-=-. The machine is supplied with a linear tape, (potentially) infinite in both directions (say to the left and right). The tape is divided into squares. Each square is capable of being blank, or of havi... |

407 |
Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I
- GÖDEL
- 1931
(Show Context)
Citation Context ...s lectures during the spring of 1934 at Princeton [6], Gödel generalized the notion of primitive recursive functions, which was introduced by him in his epoch-making paper on undecidable propositions =-=[5]-=-. 1 He did this by modifying a suggestion made by Jacques Herbrand (1908–1931) in a 1931 letter, to obtain the notion of general recursive functions (also known as the Herbrand-Gödel general recursive... |

285 |
An unsolvable problem of elementary number theory
- Church
- 1936
(Show Context)
Citation Context ...s that Gödel’s rejection of λ-definability as a possible “definition” of effective calculability was the main reason behind Church’s announcement of his thesis in terms of general recursive functions =-=[1]-=-. Church made his announcement at a meeting of the American Mathematical Society in New York City on April 19, 1935. Below is an excerpt from his abstract: Following a suggestion of Herbrand, but modi... |

90 |
On a notation for ordinal numbers
- Kleene
- 1938
(Show Context)
Citation Context ...54) formulated yet another equivalent version of computability [13]. However, his work was less detailed than Turing’s. Lastly we mention that partial recursive functions were introduced by Kleene in =-=[9]-=-. In [11] he also generalized the notion of Turing computable functions to partial functions and showed that a partial function is Turing computable if, and only if, it is partial recursive. The impor... |

73 |
General recursive functions of natural numbers
- Kleene
- 1936
(Show Context)
Citation Context ...inally accepted Church's thesis. 1It has to be noted that what we now call "primitive recursive" functions G"odel simply called "recursive." The term "primitive recursive" was introduced by Kleene in =-=[7]-=-. 2The Normal Form Theorem appeared in [7] and considerably simplified the notion of general recursive functions. 2sTuring's famous paper [15] appeared in 1936 (a correction to it was published in 193... |

64 |
The Undecidable
- Davis
- 1965
(Show Context)
Citation Context ...showed that the three notions of Turing computable, general recursive, and λ-definable functions coincide. On page 72 of Gödel’s “postscriptum” to his 1934 lecture notes which he prepared in 1964 for =-=[3]-=-, Gödel states: Turing’s work gives an analysis of the concept of “mechanical procedure” (alias “algorithm” or “computation procedure” or “finite combinatorial procedure”). This concept is shown to be... |

51 |
Lambda-definability and recursiveness
- Kleene
- 1936
(Show Context)
Citation Context ...equivalent to or weaker than recursiveness,” which indicates that, at the time, Church was not yet certain whether λ-definability was equivalent to general recursiveness. Kleene filled in this gap in =-=[8]-=- by showing that these two notions were indeed equivalent. Thus, in the full version of his paper [2], Church was already fully aware that the two notions of general recursiveness and λ-definability c... |

47 |
Recursive predicates and quantifiers
- Kleene
- 1943
(Show Context)
Citation Context ... been speculating, and finally definitely proposed, that the λ-definable functions are all the effectively calculable functions—what he published in [2], and which I in [11] Chapter XII (or almost in =-=[10]-=-) called “Church’s thesis”. When Church proposed this thesis, I sat down to disprove it by diagonalizing out of the class of the λ-definable functions. But, quickly realizing that the diagonalization ... |

45 |
On undecidable propositions of formal mathematical systems
- Gödel
- 1934
(Show Context)
Citation Context ...ico State University, Las Cruces, NM 88003; gbezhani@nmsu.edu. † With thanks to Joel Lucero-Bryan. 1it “as thoroughly unsatisfactory.” Instead, in his lectures during the spring of 1934 at Princeton =-=[6]-=-, Gödel generalized the notion of primitive recursive functions, which was introduced by him in his epoch-making paper on undecidable propositions [5]. 1 He did this by modifying a suggestion made by ... |

42 |
Finite combinatory processes formulation I
- Post
- 1936
(Show Context)
Citation Context ...te if we did not mention that around the same time and independently of Turing, but not of the work in Princeton, Emil Leon Post (1897–1954) formulated yet another equivalent version of computability =-=[13]-=-. However, his work was less detailed than Turing’s. Lastly we mention that partial recursive functions were introduced by Kleene in [9]. In [11] he also generalized the notion of Turing computable fu... |

23 | Why Gödel didn’t have Church’s thesis - Davis - 1982 |

18 | Computability and λ-definability - Turing - 1937 |

16 |
Origins of recursive function theory
- Kleene
- 1981
(Show Context)
Citation Context ...e theory of λ-definable functions. Church proposed to identify the effectively calculable functions with the λ-definable functions. Here is Kleene’s description of these events, taken from page 59 of =-=[12]-=-: The concept of λ-definability existed full-fledged by the fall of 1933 and was circulating among the logicians at Princeton. Church had been speculating, and finally definitely proposed, that the λ-... |

6 | Computability and -definability - Turing - 1937 |