Now, applying Gödel’s completeness theorem to this yieldsin turn:

There is a discussion of the Church-Turing thesis in my entry in the

The Church-Turing Thesis (Stanford Encyclopedia of …

The Turing-Church thesis concerns the notion of an or method in logic and mathematics. 'Effective' and its synonym 'mechanical' are terms of art in these disciplines: they do not carry their everyday meaning. A method, or procedure, M, for achieving some desired result is called 'effective' or 'mechanical' just in case

Church Turing Thesis Myth | Algorithms - Scribd

There are various equivalent formulations of the Turing-Church thesis (which is also known as Turing's thesis, Church's thesis, and the Church-Turing thesis). One formulation of the thesis is that every effective computation can be carried out by a Turing machine.

It was Turing’s work, Gödel emphasized, that enabled him togeneralize his incompleteness result of 1931 (originally directedspecifically at the formal system set out by Whitehead and Russell intheir Principia Mathematica) to “everyconsistent formal system containing a certain amount of finitarynumber theory” (Gödel in Davis 1965: 71).

History of the Church–Turing thesis - Wikipedia

One way in which the two men’s approaches differed was thatTuring’s concerns were rather more general than Church’s,in that (as indicated previously) Church considered only functions ofpositive integers, whereas Turing described his work as encompassing“computable functions of an integral variable or a real orcomputable variable, computable predicates, and so forth” (1936:58). Turing intended to pursue the theory of computable functions of areal variable in a subsequent paper, but in fact did not do so.

Classical physics and the Church--Turing Thesis

A significant recent contribution to the area has been made by Kripke(2013). A conventional view of the status of the Church-Turing thesisis that, while “very considerable intuitive evidence” hasamassed for the thesis, the thesis is “not a precise enoughissue to be itself susceptible to mathematical treatment”, asKripke puts it (2013: 77). Kleene gave an early expression of this nowconventional view:

On the Church-Turing thesis | Request PDF

In order to understand these assertions exactly as Turing intended them it is necessary to bear in mind that when he uses the words 'computer', 'computable' and 'computation' he employs them as pertaining to . In 1936 'computers' were human clerks who worked in accordance with effective methods. These human computers did the sort of calculations nowadays carried out by computing machines, and many thousands of them were employed in commerce, government, and research establishments. The computable numbers and the computable functions are the numbers and functions that can be calculated by human computers (idealised to the extent of living forever and having access to unlimited quantities of paper and pencils).

The Church-Turing Thesis: Breaking the Myth | …

So Turing’s and Church’s theses are equivalent. We shallusually refer to them both as Church’s thesis, or inconnection with that one of its … versions which deals with‘Turing machines’ as the Church-Turing thesis.(Kleene 1967: 232)

The physical Church-Turing thesis and the ..

Rejecting the conventional view, Kripke suggests that, on thecontrary, the Church-Turing thesis is susceptible to mathematicalproof. Furthermore he canvasses the idea that Turing himself sketchedan argument that serves to prove the thesis.