Godel Incompleteness Theorems: What They Say and Their Implications

1. A survey of proofs for Godelincompletness theorems PayamSeraji IPM-Isfahan branch, Ordibehesht 1396

2. What Godelincompleteness theorems say?

3. On formalyundecidable propositions of principamathematica 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. Why halting problem is hard? 1- input(x) 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. 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