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Details
TURING, Alan. 'Systems of logic based on ordinals.' Offprint from: Proceedings of the London Mathematical Society, ser. 2, vol. 45. London: 1939. 8° (256 x 176mm). 68pp., 161-228. Original olive-green wrappers, stapled (worn, lacking rear wrapper, front wrapper detached and chipped). Provenance: R.O. Gandy (front wrapper inscribed in pencil 'with corrections ROG', and some pencil marginalia).
OFFPRINT OF TURING'S DOCTORAL DISSERTATION, one of Turing's major works, in which he investigates ordinal logic and Gödel's theorem. 'In a "complete ordinal logic", any theorem in arithmetic could be proved by a mixture of mechanical reasoning, and steps of "intuition". In this way [Turing] hoped to bring the Godel incompleteness under some kind of control. But he regarded the results as disappointingly negative. "Complete logics" did exist, but they suffered from the defect that one could not count the number of intuitive steps that were necessary to prove any particular theorem ... This introduced the idea of relative computability, or relative unsolvability, which opened up a new field in mathematical logic' (Hodges, p.143).
OFFPRINT OF TURING'S DOCTORAL DISSERTATION, one of Turing's major works, in which he investigates ordinal logic and Gödel's theorem. 'In a "complete ordinal logic", any theorem in arithmetic could be proved by a mixture of mechanical reasoning, and steps of "intuition". In this way [Turing] hoped to bring the Godel incompleteness under some kind of control. But he regarded the results as disappointingly negative. "Complete logics" did exist, but they suffered from the defect that one could not count the number of intuitive steps that were necessary to prove any particular theorem ... This introduced the idea of relative computability, or relative unsolvability, which opened up a new field in mathematical logic' (Hodges, p.143).
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