slides gödels incompleteness theorems

8/9/2019 Slides Gdels Incompleteness Theorems

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Gdels FirstIncompletenessTheorem

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Gdels SecondIncompletenessTheorem

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Do mathematical theorems like Gdels

show that computers are intrinsically

limited?

Bas van Gijzel

May 17, 2010
http://goforward/http://find/http://goback/


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Slides and talk in English.
http://find/http://goback/


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Gdels First Incompleteness Theorem

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Gdels Second 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



Axioms: ( ) ( ( )) (( ) ( )) ( ) ( )

Inference rule(s):

Modus ponens: ,


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Formal system



Axioms: ( ) ( ( )) (( ) ( )) ( ) ( )

Inference rule(s):

Modus ponens:

,


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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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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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S


C i f f l


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S


C i t f f l t


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S


(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



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can be proved so.





Maximal consistent



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can be proved so.





Maximal consistent



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can be proved so.





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





enumerable.

Incompleteness


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Incompleteness





enumerable.

Incompleteness


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Incompleteness





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?


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

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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?



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IncompletenessTheorem

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( )

Are there unprovable truths?

S A ? SA A !



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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)



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Lucas argument:




true.







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Lucas argument:




true.







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Lucas argument:




true.







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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.



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true!

bible.

law.




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true!

bible.

law.




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true!

bible.

law.


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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?


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?


Certain amount of arithmetic.

Such a system cannot prove its own consistency.

Outline


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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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Do mathematical theorems like Gdels show that

computers are intrinsically limited?

Do mathematical theorems like Gdels show



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p y

Do mathematical theorems like Gdels show that

computers are intrinsically limited?

No!

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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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Do we even care?


What we can ask(1)


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Do we even care?


What we can ask(2)


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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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consequences?

What we can ask(2)


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consequences?

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