Payam Seraji IPM-Isfahan branch, Ordibehesht 1396 A survey of proofs for Godel incompletness theorems Payam Seraji IPM-Isfahan branch, Ordibehesht 1396

What Godel incompleteness theorems say?

On formaly undecidable propositions of principa mathematica and related systems I

Godel proved that for every theory T which has a minimum abilities and Also is , there is a proposition G aboat natural numbers such that : and We say that G is udecidable Or independet of theory T

A theory is the set of logical consequenses of a finite or recursively enumerable set of axioms :

Omega consistency A theory T is if : Then :

Ordinary consistency :If Then : Or equivalenty :

Minimum abilities T proves the following sentences : Q (Robinson’ Arithmetic)

Godel’s Proof….

Coding syntax by natural numbers

How to code a formula :

A function f is reperesentable if there is a formula A(x,y) such that :

For example function f(x)=2x can be represented by the formula :

Godel showed that every primitive recursive function is representable In the theory. Primitive recursive functions are a natural generalization of ordinary concept of functions Defined By recursion , like :

: x is godel number of the formula with the godel number y Godel showed that proof(x,y) is primitive recursive

Up to this point Godel have done what is natural if one wants to state consistency Of theory as a sentence about natural numbers ,

but now he proved a wonderful lemma. . . .

Fixed point lemma For every formula A(x) with x as the only free variable there is a sentence B such that :

By applying fixed point lemma to predicate Pr we have a sentence G such that :

Rosser’s Proof

Recursion theoretic approach There is no theory that proves all true arithmetical sentences There is no algorithm to produce all true arithmetical sentences

Mathematical definition of algorithm : Turing machine

Countable set : Uncountable : :If (0,1) is countable

Unsolvability of Halting Problem There is no algorithm to decide if an algorithm halts on a given input Halting algorithm

If all true statements of the form are Provable then the following algorithm solves the Halting problem :

Then there is sentence of the form : which is true but not provable in theory T First recursion theoretic form of Godel ‘s incompleteness theorem (Church & Turing 1936)

provable TRUE

1-input(x) 2-y:=x*x 3-print(y) Why halting problem is hard? 2- if x>0 then print Yes 3-if x=0 then goto 4 4- goto 4 1-input(x) 2-y:=x*x 3-print(y)

But every Pi-1 formula in the language of arithmetic can be seen as a Halting problem , for example the Goldbach conjecture as the following algorithm :

1-i:=4 2- check all pairs (m,n) such that m and n are both prime and smaller than i . if there was no m and n such that i=m+n then halt 3- i:=i+2 and goto 2

Goldbach conjecture is true

A set D is recursively enumerable if It is domain of a computable function It is range of a computable function

A is r.e. then there is a such that A is domain of

Recursively inseperable sets

For sufficiently strong theory T : Godel’s second incompleteness theorem For sufficiently strong theory T :

Iterative construction :

Modal logics : Language of propositional logic + ◊ □

: Necessarily p : possibly p

Gentzen question: Is G provable? Is it true ?

Lob’s theorem If then If then

Reflection principle for T (for all sentences p in the language of T)

And it contradicts the consistency of T Lob’s theorem easily implies G2 If T proves : Then by Lob’s theorem : And it contradicts the consistency of T

( (

Modal logic GL Axioms : : Rules of inference

Kripke semantic for GL : A formula s is provable in GL if it is valid in every transitive And conversely well-founded kripke model

Solovay’s completeness theorem for GL (for every trnslation From language GL To language of arithmetic)

Modal logic GLS All theorems of GL

Hilbert’s tenth problem and Matiyasevich theorem Hilbert’s 10th problem : find a algorithm to decide if A polynomial diophantie equation P(x1,…xn)=0 has Answer. Matyasevich theorem : there is no such algorithm

Every r.e. set can be represented by a diophantine equation : For every theory T with properties like before, There is a polynomial diophantine Equation That has no answer but this fact is not provable in T

Information theoretic proof of G.Chaitin Kolmogorov complexity (Algorthmic entropy)

W1= W2= W3=

1-for i:=1 to 20 do print(10); end do; W1= Is in fact the first 40 binary digits of Then it can be generated by a rilatively Short program W2= W3= Is the result of coin tossing and ,by probability Near one, there is no a program to generate it With length less than the sequence itself

Chaitin incompleteness theorem says that Kolmogorov complexity Of random Sequences almost equal the length of the sequence, but if the length of sequence is sufficiently large this fact is unprovable

Theorem (Chaitin 1972): if T is a r.e. and ω-consistent theory containitg the language of Arithmetic and contains Q (Robinson’s arithmetic), Then for sufficiently large C every true Sentence of the form K(α)>C is not provable Or disprovable in T

Relation between Kolmogorov complexity and compression of files If F is a compression program then for every file α :

Berry paradox and Boolos’s proof Some sentences in English define a number, for Example the sentence : The natural number that is both even and prime Defines number 2 and the sentence: The largest natural number which has 2 digits in decimal representation Defines the number 99

Now consider the following sentence : The least number not definable by a sentence which has at most twenty words plus one

If A(x) is a formula in the language of arithmetic, we say A(x) ‘defines’ Number n if : Using this definition Boolos formalized the sentence of Berry paradox in the language Of arithmetic and proved that it is true but not provable in theory (assuming T has Properties like maintained in Godel theorem (including )

My work : 1-Proving Rosser theorem using Chaitin’s Method 2- proving that in none of Chaitin and Boolos Proof, the undecidable sentence is not Computable from description of the theory 3- Rosserized Halting

Thank you