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

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

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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 IncompletenessFor 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 ??

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