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Turing and Ordinal Logic 14 th September, 2012 Turing and Ordinal Logic (Or “To Infinity and Beyond …”) James Harland School of Computer Science & IT, RMIT University [email protected]

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Page 1: Turing and Ordinal Logic (Or “To Infinity and Beyond …”) · 2020. 1. 20. · Turing and Ordinal Logic 14th September, 2012 Turing and Ordinal Logic (Or “To Infinity and Beyond

Turing and Ordinal Logic 14th September, 2012

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

James Harland School of Computer Science & IT, RMIT University

[email protected]

Page 2: Turing and Ordinal Logic (Or “To Infinity and Beyond …”) · 2020. 1. 20. · Turing and Ordinal Logic 14th September, 2012 Turing and Ordinal Logic (Or “To Infinity and Beyond

Turing at Princeton

Turing and Ordinal Logic 14th September, 2012

  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

Page 3: Turing and Ordinal Logic (Or “To Infinity and Beyond …”) · 2020. 1. 20. · Turing and Ordinal Logic 14th September, 2012 Turing and Ordinal Logic (Or “To Infinity and Beyond

“Overcoming” Gödel’s Incompleteness

Turing and Ordinal Logic 14th September, 2012

For any logic L1: A1 is true but unprovable in L1 L2 is L1 + A1

A2 is true but unprovable in L2 L3 is L2 + A2

A3 is 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)

Page 4: Turing and Ordinal Logic (Or “To Infinity and Beyond …”) · 2020. 1. 20. · Turing and Ordinal Logic 14th September, 2012 Turing and Ordinal Logic (Or “To Infinity and Beyond

Ordinal Logics

Turing and Ordinal Logic 14th September, 2012

  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 Π0

1 statements (!!)   Trades ‘mechanical proofs’ for completeness   Isolates where the ‘non-mechanical’ part is

Page 5: Turing and Ordinal Logic (Or “To Infinity and Beyond …”) · 2020. 1. 20. · Turing and Ordinal Logic 14th September, 2012 Turing and Ordinal Logic (Or “To Infinity and Beyond

Oracles

Turing and Ordinal Logic 14th September, 2012

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

Page 6: Turing and Ordinal Logic (Or “To Infinity and Beyond …”) · 2020. 1. 20. · Turing and Ordinal Logic 14th September, 2012 Turing and Ordinal Logic (Or “To Infinity and Beyond

Formalising ‘Intuition’

Turing and Ordinal Logic 14th September, 2012

  Turing divided mathematical reasoning into ingenuity and intuition

  Ordinal logics formalise this distinction via ordinal notation or ‘oracle’

  Extension of work on mechanising human reasoning

  Turing’s most ‘mainstream’ work ??

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