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Church thesis

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A principle according to which the class of functions computable by means of algorithms in the broad intuitive sense cf. Algorithm , coincides with the class of partial recursive functions. Church' thesis is this fact of nature, which is confirmed by the experience accumulated in mathematics throughout its history. All known examples of algorithms in mathematics satisfy it. The thesis was first formulated by A.
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Christian Church Thesis Statement

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Church–Turing thesis - Wikipedia

In computability theory , the Church—Turing thesis also known as computability thesis , [1] the Turing—Church thesis , [2] the Church—Turing conjecture , Church's thesis , Church's conjecture , and Turing's thesis is a hypothesis about the nature of computable functions. It states that a function on the natural numbers can be calculated by an effective method if and only if it is computable by a Turing machine. Before the precise definition of computable function, mathematicians often used the informal term effectively calculable to describe functions that are computable by paper-and-pencil methods. In the s, several independent attempts were made to formalize the notion of computability :. This has led mathematicians and computer scientists to believe that the concept of computability is accurately characterized by these three equivalent processes. Other formal attempts to characterize computability have subsequently strengthened this belief see below.
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Church–Turing thesis

This means that every element belonging to the former class is also a member of the latter class and reversely. Clearly, CT generates an extensional co-extensiveness of effective computability and partial recursivity. Since we have no mathematical tasks, the exact definition of recursive functions and their properties is not relevant here. On the other hand, we want to stress the property of being effective computable, which plays a basic role in philosophical thinking about CT.
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In constructive mathematics , Church's thesis CT is an axiom stating that all total functions are computable. The axiom takes its name from the Church—Turing thesis , [ citation needed ] which states that every effectively calculable function is a computable function , but the constructivist version is much stronger, claiming that every function is computable. The axiom CT is incompatible with classical logic in sufficiently strong systems. For example, Heyting arithmetic HA with CT as an addition axiom is able to disprove some instances of the law of the excluded middle. However, Heyting arithmetic is equiconsistent with Peano arithmetic PA as well as with Heyting arithmetic plus Church's thesis.
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