Gödel's Incompleteness Theorem! Dr. Philip Cannata 1

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1 Gödel's Incompleteness Theorem! Dr. Philip Cannata 1

2 Gödel's Incompleteness Theorems see Delong pages, Gödel showed that any system rich enough to express primitive recursive arithmetic (i.e., contains primitive recursive arithmetic as a subset of itself) either proves sentences which are false or it leaves unproved sentences which are true in very rough outline this is the reasoning and statement of Gödel's first incompleteness theorem. [ DeLong page, 162] Wikipedia - The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (e.g., a computer program, but it could be any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic). For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that such a system cannot demonstrate its own consistency. Dr. Philip Cannata 2

3 Gödel Numbering => V ( ) x y z = +. x 1 y 1 z 1 z 2 1 = (0) = 2 11 x 3 1 x 5 13 x 7 3 The following proof would be a sequence of sequences of symbols which would correspond to a single Gödel number (2 g1 x 3 g2 x 5 g3 ) which is the proof of = 3. See DeLong page 167 for another example g1 (0 0 0 ) g2 (0 0 (0 0 x) ) g3 (0 0 (0 ) ) => (0 0 0 ) Dr. Philip Cannata 3

4 Proof in primitive recursive arithmetic xpy (2 g1 x 3 g2 x 5 g3 ) P (2 + 1 = 3) Formula provable in A (see page 127 for the definition of A) Prov(x) = ( y)(ypx) Primitive Recursive Substitution Function Sb(x y z ) (let ((y z)) x) Pages in DeLong Universal Generalization xgeny Example: x + 0 = x ( x) x + 0 = x is 17Genc1 the first equation on page 167 is c1 Dr. Philip Cannata 4

5 Correspondence Lemma: Page 171 in DeLong For every primitive recursive relation P(x 1,,x n ) (e.g., xpy) there exists (in A) a formula (with Gödel number r) which contains free variables with Godel numbers g 1,,g n such that for all x 1,,x n : P(x 1,,x n ) -> Prov(let ((17 Nml(x 1 )) (19 Nml(x 2 )) (g n Nml(x n ))) r)) not P(x 1,,x n ) -> Prov(Neg(let ((17 Nml(x 1 )) (19 Nml(x 2 )) (g n Nml(x n ))) r))) where 17 is x and 19 is y Example: Consider = 3 x + y = z is a primitive recursive relation ((x) + (y)) = (z) is the corresponding formula in A such that = 0 is provable Dr. Philip Cannata 5

6 not xp(let ((19 Nml(y))) y) -> Prov(let ((17 Nml(x))(19 Nml(y))) q)) xp(let ((19 Nml(y))) y) -> Prov(Neg(let ((17 Nml(x))(19 Nml(y))) q))) not 0P(let ((19 Nml(p))) p) -> Prov(let ((17 Nml(0))(19 Nml(p))) q)) not 1P(let ((19 Nml(p))) p) -> Prov(let ((17 Nml(1))(19 Nml(p))) q)) not 2P(let ((19 Nml(p))) p) -> Prov(let ((17 Nml(2))(19 Nml(p))) q))... Dr. Philip Cannata 6

7 G A is ( x) not xp(let ((19 Nml(y))) y) G is formula in A that has Gödel number (let ((19 Nml(y))) y) G M is G is not provable in A. Case 1: Suppose G is provable in A, then G A is false and there is an n such that np(let ((19 Nml(y))) y), but from 2) on the previous page this would mean Prov(Neg(let ((17 Nml(n))(19 Nml(y))) q))) However, if G is provable, it also means that Prov((let ((17 Nml(n))(19 Nml(y))) q))) (see 5) on the next page) Therefore, A is inconsistent or G is not provable in A. I.e., G A is TRUE, and G is not provable if A is consistent. Dr. Philip Cannata 7

8 Case 2: Suppose not G is provable in A, Then G A is true which implies ( x) Prov(let ((17 Nml(x))(19 Nml(p))) q) Dr. Philip Cannata 8

9 G A is TRUE if A is consistent, but under the interpretation of A as primitive recursive arithmetic, G A = G. Therefore G is TRUE and not provable. Dr. Philip Cannata 9

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14 Gödel's Incompleteness Theorem If proof is a proof of statement then P is True. If you have a statement g with variable x and if, when you substitute g for x, you produce statement then Q is True. not P(proof, statement) && Q(x, statement) = g A recursive notion. not P(proof, statement) && Q(g, statement) = s Let g be the Gödel number for this statement, But now science, spurred on by its powerful delusion, hurtles inexorably towards its limits where the optimism hidden in the essence of logic founders. For the periphery of the circle of science has an infinite number of points and while there is no telling yet how the circle could ever be fully surveyed, the noble and gifted man, before he has reached the middle of his life, still inevitably encounters such peripheral limit points and finds himself staring into an impenetrable darkness. If he at that moment sees to his horror how in these limits logic coils around itself and finally bites its own tail - then the new form of knowledge breaks through, tragic knowledge, which in order to be tolerated, needs art as a protection and remedy. Friedrich Nietzsche ( ) The Birth of Tragedy Let s be the Gödel number for this statement but by the definition of Q that means statement is s. not P(proof, s) && Q(g, s) - I am a statement that is not provable. à There are Predicate Logic Statements that are True that can t be proved True (Incompleteness) and/or there are Predicate Logic Statements that can be proved True that are actually False (à Inconsistent Axioms or Unsound inference rules). i.e., If Gödel's statement is true, then it is a example of something that is true for which there is no proof. If Gödel's statement is false, then it has a proof and that proof proves the false Gödel statement true. Dr. Philip Cannata 14

15 Gödel's Incompleteness Theorem I am a statement that is not provable. à There are Predicate Logic Statements that are True that can t be proved True (Incompleteness) and/or there are Predicate Logic Statements that can be proved True that are actually False (à Inconsistent Axioms or Unsound inference rules). i.e., If Gödel's statement is true, then it is a example of something that is true for which there is no proof. If Gödel's statement is false, then it has a proof and that proof proves the false Gödel statement true. Logic/Math/CS Physics Theology Philosophy Unsound Superposition Consubstantial F S T Opposite is Excluded Middle ~p or p P L W Trace of Finite Dr. Philip Cannata The One 15 F G S H Plotinus Plato The ONE Is nothing else but The ONE, it can t even be finite. The Forms (e.g. Justice) Self Other

16 Good Books to Have for a Happy Life J From Frege to Gödel: My Favorite Dr. Philip Cannata 16

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