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# Turing and Ordinal Logic (Or “To Infinity and Beyond …”) - PowerPoint PPT Presentation

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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14th September, 2012

James Harland

School of Computer Science & IT, RMIT University

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

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

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)

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

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

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 ??