Godels First Incompleteness Theorem
- 1. Uncertainty Trumps Determinism: Godel’s Incompleteness Theorem’s and Hilbert’s Program BAE Technical Seminar
- 2. David Hilbert Kurt Godel Bertrand Russell Alfred Whitehead Georg Cantor The Founders of the Modern Mathematical Foundation
- 3. Platonists regard numbers as independently existing objects Formalists regard mathematics as nothing more nor less than language Intuitionists regard numbers as being unique inventions of the mind Grundlagenkrise der Mathematik
- 4. Avoiding Paradox or Embracing it? Russell’s Paradox: R = {A | A ∉ A} Liar’s Paradox: All Cretans are liars. I am a Cretan. Godel’s “sentence”: This statement is true but cannot be proved.
- 5. Hilbert’s Second Problem (1900) “… with regard to the axioms: To prove that they are not contradictory, that is, that a definite number of logical steps based upon them can never lead to contradictory results. In geometry, the proof of the compatibility of the axioms can be effected by constructing a suitable field of numbers, such that analogous relations between the numbers of this field correspond to the geometrical axioms. ... On the other hand a direct method is needed for the proof of the compatibility of the arithmetical axioms.”
- 6. Hilbert’s Program (circa 1921) A formalization of all mathematics; in other words all mathematical statements should be written in a precise formal language, and manipulated according to well defined rules
- 7. Hilbert’s Program Formalization: the principle that all mathematical statements should be written in a precise formal language, and manipulated according to well defined rules. Consistency: a proof that no contradiction can be obtained in the formalism of mathematics. This consistency proof should preferably use only "finitistic" reasoning about finite mathematical objects.
- 8. Hilbert’s Program Completeness: a proof that all true mathematical statements can be proved in the formalism. Decidability: there should be an algorithm for deciding the truth or falsity of any mathematical statement.
- 9. Godel’s Answer to Hilbert’s Program (1931) Any effectively generated theory T capable of expressing elementary arithmetic cannot be both consistent and complete . In particular, T generates a statement which is true but unprovable within T. For any formal recursively enumerable theory T including basic arithmetical truths …, T includes a statement of its own consistency if and only if T is inconsistent.
- 10. A Simple Language for a Godelian Problem – Is Printable Consider a machine that prints “sentences” composed from the following symbols: ~ P N ( ) Define a sentence as one of the following four forms: P(X) PN(X) ~P(X) ~PN(X) … and define the norm of X as X(X)
- 11. Is Printable Define P(X) is true iff X is printable. Define PN(X) is true iff the norm of X is printable (iff X(X) is printable) Define ~PN(X) is true iff the norm of X is not printable Assume the machine is accurate , i.e. any sentence it prints is true.
- 12. Is Printable Can our machine print every true sentence?
- 13. Is Printable Can our machine print every true sentence? ~PN(~PN)
- 14. Is Printable Can our machine print every true sentence? ~PN(~PN) This says the norm of ~PN is not printable.
- 15. Is Printable Can our machine print every true sentence? ~PN(~PN) This says the norm of ~PN is not printable. But the norm of ~PN is ~PN(~PN) !
- 16. Unprovable, Unrefutable, Undecidable ~PN(~PN) This sentence is true iff it is not printable, so either it is true and not printable or printable and not true. The latter contradicts the hypothesis that the machine is accurate , so it must be that the sentence is true and not printable. What about the sentence PN(~PN)? It’s false since it’s negation is true, therefore not printable Substitute “provable” for “printable” and we’ve found an undecidable assertion in this language.
- 17. A Simple Godel Numbering Assign to the symbols ~ P N the numbers 10, 100, 1000, and in any expression, replace the symbol by its base 10 numeral expression. Define the norm of an expression to be that expression followed by its Godel number … e.g. PNP1001000100 From our previous game, the undecidable expression in this notation is ~PN101001000
- 18. An Abstract Language, L , for Godel logic A denumerable set E of expressions of L S ⊂ E , called the sentences of L P ⊂ S , called the provable sentences of L R ⊂ S , called the refutable sentences of L H ⊂ E , called the predicates of L A function φ : E x N -> E (if E ∈ H , E is a sentence) T ⊂ S called the true sentences of L Define: H(n) expresses a set A ⊂ N by … H(n) ∈ T ↔ n ∈ A
- 19. We call a system L correct if every provable sentence is true and every refutable sentence is not true. So ( P ⋂ T) = P and ( R ⋂ T) = ∅
- 20. We call a system L correct if every provable sentence is true and every refutable sentence is not true. So ( P ⋂ T) = P and ( R ⋂ T) = ∅ Does P ⋂ T = T as well?
- 21. We call a system L correct if every provable sentence is true and every refutable sentence is not true. So ( P ⋂ T) = P and ( R ⋂ T) = ∅ Does P ⋂ T = T as well? NO!
- 22. We call a system L correct if every provable sentence is true and every refutable sentence is not true. So ( P ⋂ T) = P and ( R ⋂ T) = ∅ Does P ⋂ T = T as well ? NO! If L is correct, under what conditions must it contain a true sentence not provable in L ?
- 23. Godel Numbering and Diagonalization Let g: E -> N be 1-1 function assigning to each expression E its Godel number , n, i.e. g(E)=n Let E n be that expression whose Godel number is n, i.e. … ∀ n(n is a Godel number ↔ (g(E n ) = n)) Define: The diagonalization of E n to be E n (n). If E n is a predicate (and therefore a sentence) then it’s true iff it’s satisfied by its own Godel number … (… by H(n) ∈ T ↔ n ∈ A …)
- 24. Why Diagonal ?
- 25. Why Diagonal ? s 0 = (0, 0, 0, 0, 0, 0, 0, ...) s 1 = (1, 1, 1, 1, 1, 1, 1, ...) s 2 = (0, 1, 0, 1, 0, 1, 0, ...) s 3 = (1, 0, 1, 0, 1, 0, 1, ...) s 4 = (1, 1, 0, 1, 0, 1, 1, ...) s 5 = (0, 0, 1, 1, 0, 1, 1, ...) s 6 = (1, 0, 0, 0, 1, 0, 0, ...) . . . . . . . . . . . . . . . . . . . . . s n = ( s n,0 , s n,1 , …, s n,m , s n,n ,… ) Georg Cantor published his diagonal argument in 1891. It’s a method for demonstrated that there are undenumerable sets of numbers.
- 26. Why Diagonal ? s 0 = ( 0 , 0, 0, 0, 0, 0, 0, ...) s 1 = (1, 1 , 1, 1, 1, 1, 1, ...) s 2 = (0, 1, 0 , 1, 0, 1, 0, ...) s 3 = (1, 0, 1, 0 , 1, 0, 1, ...) s 4 = (1, 1, 0, 1, 0 , 1, 1, ...) s 5 = (0, 0, 1, 1, 0, 1 , 1, ...) s 6 = (1, 0, 0, 0, 1, 0, 0 , ...) . . . . . . . . . . . . . . . . . . . . . s n = ( s n,0 , s n,1 , …, s n,m , s n,n … ) It’s possible to construct a sequence that is different in the from all other sequences in the matrix, e.g.: s ω+1 = (1, 0, 1, 1, 1, 0, 1, …)
- 27. Why Diagonal ? s 0 = ( 0 , 0, 0, 0, 0, 0, 0, ...) s 1 = (1, 1 , 1, 1, 1, 1, 1, ...) s 2 = (0, 1, 0 , 1, 0, 1, 0, ...) s 3 = (1, 0, 1, 0 , 1, 0, 1, ...) s 4 = (1, 1, 0, 1, 0 , 1, 1, ...) s 5 = (0, 0, 1, 1, 0, 1 , 1, ...) s 6 = (1, 0, 0, 0, 1, 0, 0 , ...) . . . . . . . . . . . . . . . . . . . . . s n = ( s n,0 , s n,1 , …, s n,m , s n,n … ) *This proves that there are uncountable (undenumerable) sets of numbers (and in fact proves that the Reals are undenumerable) It’s possible to construct a sequence that is different in the from all other sequences in the matrix, e.g.: s ω+1 = (1, 0, 1, 1, 1, 0, 1, …)*
- 28. The Diagonalization Function For any n, we let d(n) be the Godel number of the expression E n (n) and call it the diagonal function of the system. Define the set A* to be the set of all numbers, n, such that d(n) ∈ A, i.e. … n ∈ A* ↔ d(n) ∈ A Define c A, the complement of A in N with the usual meaning of complementation and let P be the set of Godel numbers of the provable sentences of L
- 29. Theorem 1 If the set c P* is expressible in L and L is correct, then there is a true sentence of L not provable in L. By hypothesis, L is correct and c P* is expressible. Let H be a predicate expressing c P* in L with Godel number h. Let G=H(h) (the diagonalization of H) . We’ll show that G is true but unprovable in L .
- 30. Proof of Theorem 1 Since H expresses c P*, then for any n, H(n) is true iff n ∈ c P*. In particular, H(h) is true so H(h) is true iff h ∈ c P*. And .. h ∈ c P* ↔ d(h) ∈ c P ↔ d(h) ∉ P But d(h) is the Godel number of G=H(h) and so d(h) ∈ P ↔ G is provable. So either G is true and not provable or it is untrue and provable. By correctness hypothesis, it must be true but not provable. Q.E.D.
- 31. We have assumed two things – the correctness of L and the expressibility of c P*. For a correct language L the following conditions hold: G1: A expressible implies A* expressible G2: A expressible implies c A expressible G3: The set P (of provable sentences) is expressible
- 32. Call an expression E n a Godel Sentence for a number set A is either E n is true and its Godel number lies in A or it’s false and its Godel number lies in c A. E n is a Godel sentence for A iff this condition holds: E n ∈ T ↔ n ∈ A
- 33. The Set T , Expressibility The Diagonal Lemma: For any set A, let G A be its Godel sentence For any set A, A* expressible in L -> ∃G A If L satisfies G 1 , then for any set A expressible in L , ∃G A
- 34. Let T be the set of Godel numbers of the true sentences of L Tarski’s Undefinability Theorem: The set c T* is not expressible in L If G 1 holds, c T is not expressible in L If G 1 & G 2 hold, T is not expressible in L
- 35. Decidability Completeness: We say a sentence (in L ) is decidable if it is either provable or refutable (in L ); we say a system is complete if every sentence is decidable. Consistency: We say a system is consistent if no sentence is both provable and refutable (( P ⋂ R) = ∅).
- 36. Incompleteness We can restate our Theorem 1 as follows: If L is correct and the set c P* is expressible in L , then L is incomplete. This follows from our Godel sentence G being true but unprovable. Since it’s true, it’s also not refutable (because L is correct). Hence, G is an undecidable sentence.
- 37. Refutability We’ve not dealt with the set R , the set of Godel numbers of the refutable sentences, i.e. with sentences of the form “I am refutable” as opposed to “I am not provable.” Presumably, we can show that such sentences are also undecidable.
- 38. Alternative to Theorem 1 If L is correct and the set R* is expressible in L , then L is incomplete. The proof mirrors that of its sibling theorem.
- 39. Expressibility as a Metaphor for Arithmetic Godel’s proof was over the axioms of Peano Arithmetic (P.A.). Assume all statements over N For any x, x=x For any x, y, if x=y then y=x For any x, y, z, if x=y and y=z then x=z For any a, b, a ∈ N; a = b -> b ∈ N 0 ∈ N For every n ∈ N , S(n) ∈ N For every n ∈ N , S(n) ≠ 0 For {m, n} ∈ N, S(m) = S(n) -> n=m If 0 ∈ K and for all n ∈ N, n ∈ K then S(n) ∈ K
- 40. We can now define operations over L as arithmetic (strictly formulable from the P.A. axioms) and we find familiar postulates from the general theory: Diagonalization: n ∈ A* ↔ d(n) ∈ A If A is arithmetic, so is A* If A is arithmetic, so is c A For every arithmetic set A ∃a Godel Sentence The set T of Godel numbers of the true arithmetic sentences is not arithmetic
- 41. Correctness versus Consistency Define ω-inconsistency : An axiom system S is said to ω-inconsistent if there is a formula, F(w) such that ∃ w F( w) is provable yet all the sentences F(0), F(1), F(2), ... are refutable. A system that is not ω-inconsistent is ω-consistent.
- 42. Godel’s 2 nd Incompleteness Theorem For any formal recursively enumerable theory T including basic arithmetical truths …, T includes a statement of its own consistency if and only if T is inconsistent. More forcefully, this theorem says that any such system that can prove it’s own consistency is inconsistent!
- 43. How’d it Work out for Hilbert’s Program? It is not possible to formalize all of mathematics There is no complete, consistent extension even of P.A. – most “interesting” systems are incomplete A theory as complex as P.A. cannot demonstrate its own consistency There is no algorithm to decide truth or provability of systems considered here (Hilbert’s Entscheidungsproblem)