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

2. What Godel incompleteness theorems say?

3. On formaly undecidable propositions of principa mathematica and related
systems I

4. 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

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

6. Omega consistency
A theory T is if :
Then :

7. Ordinary consistency
:If
Then :
Or equivalenty :

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

9. Godel’s Proof….

10. Coding syntax by natural numbers

11. How to code a formula :

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

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

14. 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 :

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

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

18. but now he proved a wonderful lemma. . . .

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

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

23. Rosser’s Proof

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

25. Mathematical definition of algorithm
: Turing machine

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

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

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

30. 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)

31. provable
TRUE

32. 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)

33. 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 :

34. 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

35. Goldbach conjecture is true

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

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

40. Recursively inseperable sets

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

46. Iterative construction :

47. Modal logics
: Language of propositional logic +
◊ □

48. : Necessarily p
: possibly p

51. Gentzen question:
Is G provable? Is it true ?

52. Lob’s theorem If
then If
then

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

54. 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

55. ( (

56. Modal logic GL
Axioms :
: Rules of inference

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

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

62. Modal logic GLS
All theorems of GL

63. 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

64. 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

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

66. W1=
W2=
W3=

67. 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

68. 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

69. 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

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

71. 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

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

73. 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 )

74. 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

75. Thank you