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Do mathematical theorems like Gdels
show that computers are intrinsically
limited?
Bas van Gijzel
May 17, 2010
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Slides and talk in English.
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Outline
Gdels First Incompleteness Theorem
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Outline
Gdels First Incompleteness Theorem
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Not one but two
Not Gdels incompleteness theorem but Gdelsincompleteness theorems!
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First incompleteness theorem
The first incompleteness theorem (Gdel-Rosser).
Anyconsistentformal system S within which a
certain amount of elementary arithmetic can be
carried out is incomplete with regard to
statements of elementary arithmetic: there are
such statements which can neither be proved, nor
disproved in S.
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Important concepts
Elementary arithmetic
Formal system
Proved/disproved
Consistency of a formal system
Completeness
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Crash course in logic concepts
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Formal system
Formal system: axioms, inference rules.
Propositional logic:
Axioms: ( ) ( ( )) (( ) ( )) ( ) ( )
Inference rule(s):
Modus ponens: ,
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Formal system
Formal system: axioms, inference rules.
Propositional logic:
Axioms: ( ) ( ( )) (( ) ( )) ( ) ( )
Inference rule(s):
Modus ponens: ,
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Formal system
Formal system: axioms, inference rules.
Propositional logic:
Axioms: ( ) ( ( )) (( ) ( )) ( ) ( )
Inference rule(s):
Modus ponens:
,
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Elementary arithmetic
Intuitively at least the following:
Natural numbers: 0, 1, 2, . . . .
Using 0 and a successor function S. Addition and multiplication.
Induction principle.
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Elementary arithmetic
Intuitively at least the following:
Natural numbers: 0, 1, 2, . . . .
Using 0 and a successor function S. Addition and multiplication.
Induction principle.
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Elementary arithmetic
Intuitively at least the following:
Natural numbers: 0, 1, 2, . . . .
Using 0 and a successor function S. Addition and multiplication.
Induction principle.
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Elementary arithmetic
Intuitively at least the following:
Natural numbers: 0, 1, 2, . . . .
Using 0 and a successor function S. Addition and multiplication.
Induction principle.
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Statements of a system
A statement is a logical formula, for instance pp.
A statement that is provable in S, is denoted as S .
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Statements of a system
A statement is a logical formula, for instance pp.
A statement that is provable in S, is denoted as S .
C
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Consistency of a formal system
Intuitively: a system does not derive nonsense.
A system S is consistent iffS .
Or, a system cannot simultaneously derive S and
S
More to be said in the discussion. . .
C i f f l
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Consistency of a formal system
Intuitively: a system does not derive nonsense.
A system S is consistent iffS .
Or, a system cannot simultaneously derive S and
S
More to be said in the discussion. . .
C i f f l
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Consistency of a formal system
Intuitively: a system does not derive nonsense.
A system S is consistent iffS .
Or, a system cannot simultaneously derive S and
S
More to be said in the discussion. . .
C i t f f l t
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Consistency of a formal system
Intuitively: a system does not derive nonsense.
A system S is consistent iffS .
Or, a system cannot simultaneously derive S and
S
More to be said in the discussion. . .
(N ti ) c l t ss f th
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(Negation) completeness of a theory
Intuitively: All statements are either true or untrue and
can be proved so.
Very strong property!
System T is complete ifffor every sentence : T orT .
Not: every true formula can be proved.
(Completeness of FOL)
Maximal consistent
(Negation) completeness of a theory
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(Negation) completeness of a theory
Intuitively: All statements are either true or untrue and
can be proved so.
Very strong property!
System T is complete ifffor every sentence : T orT .
Not: every true formula can be proved.
(Completeness of FOL)
Maximal consistent
(Negation) completeness of a theory
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(Negation) completeness of a theory
Intuitively: All statements are either true or untrue and
can be proved so.
Very strong property!
System T is complete ifffor every sentence : T orT .
Not: every true formula can be proved.
(Completeness of FOL)
Maximal consistent
(Negation) completeness of a theory
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(Negation) completeness of a theory
Intuitively: All statements are either true or untrue and
can be proved so.
Very strong property!
System T is complete ifffor every sentence : T orT .
Not: every true formula can be proved.
(Completeness of FOL)
Maximal consistent
Incompleteness
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Incompleteness
System does not have good enough inference rules.
There is a sentence for T, , for which T and T .
Derivation is undecidable.
Or: sentences of the system are not recursively
enumerable.
Incompleteness
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Incompleteness
System does not have good enough inference rules.
There is a sentence for T, , for which T and T .
Derivation is undecidable.
Or: sentences of the system are not recursively
enumerable.
Incompleteness
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Incompleteness
System does not have good enough inference rules.
There is a sentence for T, , for which T and T .
Derivation is undecidable.
Or: sentences of the system are not recursively
enumerable.
Incompleteness
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Incompleteness
System does not have good enough inference rules.
There is a sentence for T, , for which T and T .
Derivation is undecidable.
Or: sentences of the system are not recursively
enumerable.
End of crash course
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End of crash course
First incompleteness theorem (again)
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First incompleteness theorem (again)
The first incompleteness theorem (Gdel-Rosser).
Anyconsistentformal system S within which a
certain amount of elementary arithmetic can be
carried out is incomplete with regard tostatements of elementary arithmetic: there are
such statements which can neither be proved, nor
disproved in S.
Summary
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y
So what does Gdels first Incompleteness Theorem say?
About axiomatised formal theories of arithmetic.
In short, arithmetical truth isnt provability in some
single axiomatisable system.
IfT is consistent: GT : T GT and T GT.
But it also holds that: ConT GT.
Summary
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y
So what does Gdels first Incompleteness Theorem say?
About axiomatised formal theories of arithmetic.
In short, arithmetical truth isnt provability in some
single axiomatisable system.
IfT is consistent: GT : T GT and T GT.
But it also holds that: ConT GT.
Outline
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Gdels First Incompleteness TheoremIntroduction
Discussion
Gdels Second Incompleteness TheoremIntroduction
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Answer to the questionMore questions
Possible questions (1)
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q ( )
Are there unprovable truths?
Possible questions (1)
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( )
Are there unprovable truths?
S A ? SA A !
Possible questions (2)
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Lucas argument:
Gdels theorem states that in any consistentsystem which is strong enough to produce simple
arithmetic there are formulas which cannot be
proved in the system, but which we can see to be
true.
We have to prove consistency for that system!
Humans cannot prove formal systems consistent in
general. Going out of the system is not possible in general.
(Goldbachs conjecture)
Possible questions (2)
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Lucas argument:
Gdels theorem states that in any consistentsystem which is strong enough to produce simple
arithmetic there are formulas which cannot be
proved in the system, but which we can see to be
true.
We have to prove consistency for that system!
Humans cannot prove formal systems consistent in
general. Going out of the system is not possible in general.
(Goldbachs conjecture)
Possible questions (2)
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Lucas argument:
Gdels theorem states that in any consistentsystem which is strong enough to produce simple
arithmetic there are formulas which cannot be
proved in the system, but which we can see to be
true.
We have to prove consistency for that system!
Humans cannot prove formal systems consistent in
general. Going out of the system is not possible in general.
(Goldbachs conjecture)
Possible questions (2)
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Lucas argument:
Gdels theorem states that in any consistentsystem which is strong enough to produce simple
arithmetic there are formulas which cannot be
proved in the system, but which we can see to be
true.
We have to prove consistency for that system!
Humans cannot prove formal systems consistent in
general. Going out of the system is not possible in general.
(Goldbachs conjecture)
(Negation) completeness of a theory
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The . . . claims to give all answers/claims to be a complete
system. By Gdels incompleteness theorems this cannot be
true!
bible.
law.
system of propositional logic.
(Negation) completeness of a theory
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The . . . claims to give all answers/claims to be a complete
system. By Gdels incompleteness theorems this cannot be
true!
bible.
law.
system of propositional logic.
(Negation) completeness of a theory
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The . . . claims to give all answers/claims to be a complete
system. By Gdels incompleteness theorems this cannot be
true!
bible.
law.
system of propositional logic.
(Negation) completeness of a theory
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The . . . claims to give all answers/claims to be a complete
system. By Gdels incompleteness theorems this cannot be
true!
bible.
law.
system of propositional logic.
Outline
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Gdels First Incompleteness TheoremIntroduction
Discussion
Gdels Second Incompleteness TheoremIntroduction
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Answer to the questionMore questions
Outline
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Gdels First Incompleteness TheoremIntroduction
Discussion
Gdels Second Incompleteness TheoremIntroduction
Discussion
Answer to the questionMore questions
Second incompleteness theorem
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The second incompleteness theorem (Gdel).
For any consistent formal system S within which a
certain amount of elementary arithmetic can becarried out, the consistency of S cannot be proved
in S itself.
Amount of elementary arithmetics
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Different amount of arithmetics!
Gdel numbering of sentences.
Amount of elementary arithmetics
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Different amount of arithmetics!
Gdel numbering of sentences.
Summary
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So what do Gdels Second Incompleteness Theorems say?
About axiomatised formal theories of arithmetic.
Certain amount of arithmetic.
Such a system cannot prove its own consistency.
Summary
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So what do Gdels Second Incompleteness Theorems say?
About axiomatised formal theories of arithmetic.
Certain amount of arithmetic.
Such a system cannot prove its own consistency.
Outline
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Gdels First Incompleteness TheoremIntroduction
Discussion
Gdels Second Incompleteness TheoremIntroduction
Discussion
Answer to the questionMore questions
Outline
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Gdels First Incompleteness TheoremIntroduction
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Gdels Second Incompleteness TheoremIntroduction
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What we can claim
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Arithmetical truth isnt provability in some single
axiomatisable system.
No super foundation of mathematics that is complete.(Principia Mathematica)
What we can claim
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Arithmetical truth isnt provability in some single
axiomatisable system.
No super foundation of mathematics that is complete.(Principia Mathematica)
What we should not claim
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By Gdels theorems I hereby pronounce. . . machines less powerful than humans.
formal systems useless. logic as an attempt to formalise AI useless.
Do mathematical theorems like Gdels show
that computers are intrinsically limited?
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Gdels First
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that computers are intrinsically limited?
Do mathematical theorems like Gdels show that
computers are intrinsically limited?
Do mathematical theorems like Gdels show
that computers are intrinsically limited?
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Gdels First
IncompletenessTheorem
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Discussion
Gdels SecondIncompletenessTheorem
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Discussion
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p y
Do mathematical theorems like Gdels show that
computers are intrinsically limited?
No!
Outline
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Gdels First
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Gdels SecondIncompletenessTheorem
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Gdels First Incompleteness TheoremIntroduction
Discussion
Gdels Second Incompleteness TheoremIntroduction
Discussion
Answer to the questionMore questions
What we can ask(1)
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Gdels First
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Are humans formal systems?
Are humans complete?
Do we even care?
An infinitude of human formal systems?
What we can ask(1)
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Gdels First
IncompletenessTheorem
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Gdels SecondIncompletenessTheorem
Introduction
Discussion
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Are humans formal systems?
Are humans complete?
Do we even care?
An infinitude of human formal systems?
What we can ask(1)
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Gdels First
IncompletenessTheorem
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Gdels SecondIncompletenessTheorem
Introduction
Discussion
Answer to thequestion
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Are humans formal systems?
Are humans complete?
Do we even care?
An infinitude of human formal systems?
What we can ask(2)
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Gdels First
IncompletenessTheorem
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Gdels SecondIncompletenessTheorem
Introduction
Discussion
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What can humans effectively calculate?
Related: Church-Turing thesis and its variants.
Are mathematics useful if they have so little practical
consequences?
What we can ask(2)
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Gdels First
IncompletenessTheorem
Introduction
Discussion
Gdels SecondIncompletenessTheorem
Introduction
Discussion
Answer to thequestion
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37
What can humans effectively calculate?
Related: Church-Turing thesis and its variants.
Are mathematics useful if they have so little practical
consequences?
What we can ask(2)
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Gdels First
IncompletenessTheorem
Introduction
Discussion
Gdels SecondIncompletenessTheorem
Introduction
Discussion
Answer to thequestion
More questions
37
What can humans effectively calculate?
Related: Church-Turing thesis and its variants.
Are mathematics useful if they have so little practical
consequences?
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