Turing's proof

Turing's proof is a proof by Alan Turing, first published in January 1937 with the title "On Computable Numbers, With an Application to the Entscheidungsproblem." It was the second proof of the assertion (Alonzo Church's proof was first) that some decision problems are "undecidable": there is no single algorithm that infallibly gives a correct "yes" or "no" answer to each instance of the problem. In his own words: "...what I shall prove is quite different from the well-known results of Gödel ... I shall now show that there is no general method which tells whether a given formula U is provable in K [Principia Mathematica]..." (Undecidable, p. 145).

Turing preceded this proof with two others. The second and third both rely on the first. All rely on his development of type-writer-like "computing machines" that obey a simple set of rules and his subsequent development of a "universal computing machine".

In 1905 Jules Richard presented this profound paradox. Alan Turing's first proof constructs this paradox with his so-called computing machine and proves that this machine cannot answer a simple question: will this machine be able to determine if any computing machine (including itself) will become trapped in an unproductive "infinite loop" (i.e. it fails to continue its computation of the diagonal number).

A succinct definition of Richard's paradox is found in Whitehead and Russell's Principia Mathematica:

Turing's proof is complicated by a large number of definitions, and confounded with what Martin Davis called "petty technical details" and "...technical details [that] are incorrect as given" (Davis's commentary in Undecidable, p. 115). Turing himself published "A correction" in 1937: "The author is indebted to P. Bernays for pointing out these errors" (Undecidable, p. 152).

Specifically, in its original form the third proof is badly marred by technical errors. And even after Bernays' suggestions and Turing's corrections, errors remained in the description of the universal machine. And confusingly, since Turing was unable to correct his original paper, some text within the body harks to Turing's flawed first effort.

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