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Due to his unwillingness to work as hard on his classical studies as on science and mathematics, Turing failed his final examinations several times, and went on to the college of his second choice, King's College, Cambridge, rather than his first choice, Trinity. He studied under G. H. Hardy, a well respected mathematician who held the Sadleirian Chair at Cambridge, then a centre for mathematical research and study, from 1931 to 1934. In 1935 he was elected a Fellow at King's College.
In his momentous paper "On Computable Numbers, with an Application to the Entscheidungsproblem" (submitted on May 28, 1936), Turing reformulated Kurt Gödel's 1931 results on the limits of proof and computation, substituting Gödel's universal arithmetics-based formal language by what are now called Turing machines, formal and simple devices. He proved that such a machine would be capable of performing any conceivable mathematical problem if it were representable as an algorithm, even if no actual Turing machine would be likely to have practical applications, being much slower than alternatives. Turing machines are to this day the central object of study in theory of computation. He went on to prove that there was no solution to the Entscheidungsproblem by first showing that the halting problem for Turing machines is uncomputable: it is not possible to algorithmically decide whether a given Turing machine will ever halt. While his proof was published subsequent to Alonzo Church's equivalent proof in respect to his lambda calculus, Turing's work is considerably more accessible and intuitive. It was also novel in its notion of a "Universal (Turing) Machine", the idea that such a machine could perform the tasks of any other machine. The paper also introduces the notion of definable numbers.
Most of 1937 and 1938 he spent at Princeton University, studying under Alonzo Church. In 1938 he obtained his Ph.D. from Princeton; his dissertation introduced the notion of hypercomputation where Turing machines are augmented with so-called oracles, allowing a study of problems that cannot be solved algorithmically.
Back in Cambridge in 1939, he attended lectures by Ludwig Wittgenstein about the foundations of mathematics. The two argued and disagreed vehemently, with Turing defending formalism and Wittgenstein arguing that mathematics is overvalued and does not discover any absolute truths.
During the World War II he was a major participant in the efforts at Bletchley Park on cracking Nazi cyphers. He contributed several mathematical insights, both to breaking the Enigma cypher machine and the Fish teletype cyphers (teletype cypher machines made by both Lorenz Electric and Siemens & Halske). The Fish insights were useful in the development of the first digital programmable electronic computer Colossus, which was designed by Max Newman and team, and built at the Post Office Research Station at Dollis Hill by a team led by Thomas Flowers in 1943. It was used to crack Fish cyphers (in particular the Lorenz machine traffic).
To help crack the Enigma machine, Turing designed the bombe, an electromechanical machine - named in recognition of the Polish-designed bomba - which could be used to eliminate large numbers of candidate Enigma keys. For each possible setting, a chain of logical deductions was implemented electrically. It was possible to detect when a contradiction had occurred and rule out that setting. Turing's bombe, with an enhancement suggested by mathematician Gordon Welchman , was the primary tool used by Allied codebreakers to read Enigma traffic.
Turing's codebreaking work was kept secret until the 1970s; not even his close friends knew about it.
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