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Turing and Ordinal Logic (Or “To Infinity and Beyond …”)

James Harland School of Computer Science & IT, RMIT University james.harland@rmit.edu.au. Turing and Ordinal Logic (Or “To Infinity and Beyond …”). Turing at Princeton. Arrived in September 1936 Worked on type theory and group theory Spent June-September 1937 back in England

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Turing and Ordinal Logic (Or “To Infinity and Beyond …”)

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  1. 14th September, 2012 James Harland School of Computer Science & IT, RMIT University james.harland@rmit.edu.au Turing and Ordinal Logic(Or “To Infinity and Beyond …”)

  2. 14th September, 2012 Turing at Princeton • Arrived in September 1936 • Worked on type theory and group theory • Spent June-September 1937 back in England • Returned to Princeton in September 1937 • Procter Fellow (on $2000 per year) (!) • Church’s PhD student (!!) • Systems of Logic Based on Ordinals completed in May, 1938 (!!!) • Returned to Cambridge in July, 1938

  3. 14th September, 2012 “Overcoming” Gödel’s Incompleteness For any logic L1: A1 is true but unprovable in L1 L2 is L1 + A1 A2is true but unprovable in L2 L3 is L2 + A2 A3is true but unprovable in L3 L4 is L3 + A3 … ``The situation is not quite so simple as is suggested by this crude argument’’ (first page of 68)

  4. 14th September, 2012 Ordinal Logics • In general, need to add an infinite number of axioms • Create ‘ordinal logics’ in the same way as ordinal numbers • Pass to the transfinite in order to obtain completeness • (… lots of technicality omitted …) • Completeness obtained (for Π01 statements) • “Proofs” now not just ‘mechanical’ • Need to recognise when a formula is an ‘ordinal formula’ • This is at least as hard as Π01 statements (!!) • Trades ‘mechanical proofs’ for completeness • Isolates where the ‘non-mechanical’ part is

  5. 14th September, 2012 Oracles • Oracle: Answer an unsolvable problem • o-machines: add this capability to Turing machines • Provides computational conception of limit ordinal process • Showed that the halting problem for o-machines cannot be solved by an o-machine • Genesis of the idea of relative computability & generalised recursion theory

  6. 14th September, 2012 Formalising ‘Intuition’ • Turing divided mathematical reasoning into ingenuity and intuition • Ordinal logics formalise this distinction viaordinal notation or ‘oracle’ • Extension of work on mechanising human reasoning • Turing’s most ‘mainstream’ work ??

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