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James Harland School of Computer Science & IT, RMIT University [email protected] 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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James harland school of computer science it rmit university james harland@rmit edu au

14th September, 2012

James Harland

School of Computer Science & IT, RMIT University

[email protected]

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


Turing at princeton

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


Overcoming g del s incompleteness

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)


Ordinal logics

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


Oracles

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


Formalising intuition

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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