Dr. Philip Cannata 1

Gödel's Incompleteness Theorem!

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Gödel's Incompleteness Theorems – see Delong pages, 165 - 180

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.

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Gödel Numbering 1 3 5 7 9 11 13 17 19 23

‘0’ ‘’’ ‘-’ ‘=>’ ‘V’ ‘(‘ ‘) ‘x’ ‘y’ ‘z’

29 31 37 41 43 47 53 … ‘=‘ ‘+’ ‘.’ ‘x1’ ‘y1’

‘z1’

‘z2’

…

1 = (0)’ = 211 x 31 x 513 x 73

The following proof would be a sequence of sequences of symbols which would correspond to a single Gödel number (2g1 x 3g2 x 5g3) which is the proof of 2 + 1 = 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’’’)

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Proof in primitive recursive arithmetic xPy (2g1 x 3g2 x 5g3) 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 yz) (let ((y z)) x)

Universal Generalization xGeny Example: x + 0 = x (∀x) x + 0 = x is 17Genc1 – the first equation on page 167 is c1

Pages 166 – 170 in DeLong

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Correspondence Lemma: For every primitive recursive relation P(x1,…,xn) (e.g., xPy) there exists (in A) a formula (with Gödel number r) which contains free variables with Godel numbers g1, …,gn such that for all x1,…,xn:

P(x1,…,xn) -> Prov(let ((17 Nml(x1)) (19 Nml(x2))…(gn Nml(xn))) r))

not P(x1,…,xn) -> Prov(Neg(let ((17 Nml(x1)) (19 Nml(x2))…(gn Nml(xn))) r))) where 17 is ‘x’ and 19 is ‘y’

Example:

Consider 1 + 2 = 3 x + y = z is a primitive recursive relation ((x) + (y)) = (z) is the corresponding formula in A such that 0’ + 0’’ = 0’’’ is provable

Page 171 in DeLong

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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)) . . .

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GA is (∀x) not xP(let ((19 Nml(y))) y)

G is formula in A that has Gödel number (let ((19 Nml(y))) y)

GM is “G is not provable in A”.

Case 1: Suppose G is provable in A, then GA 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., GA is TRUE, and G is not provable if A is consistent.

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Case 2: Suppose not G is provable in A, Then GA is true which implies (∀x) Prov(let ((17 Nml(x))(19 Nml(p))) q)

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

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not P(proof, statement) && Q(x, statement) = g

not P(proof, statement) && Q(g, statement) = 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.

Let s be the Gödel number for this statement but by the definition of Q that means “statement” is “s”.

Gödel's Incompleteness Theorem If you have a statement g with variable x and if, when

you substitute g for x, you produce “statement” then Q is True.

If “proof” is a proof of “statement” then P is True.

A recursive notion.

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 (1844 - 1900) The Birth of Tragedy

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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.

Gödel's Incompleteness Theorem

S

T F

L

WP

G

S F H

Logic/Math/CS Physics Theology Philosophy

Unsound Superposition Consubstantial

Opposite is Excluded Middle

~p or p

The ONE Is nothing else but The ONE,

it can’t even be finite.

The Forms (e.g. Justice)

Self Trace of The One

Other Finite

…

Plotinus

Plato

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Good Books to Have for a Happy Life J

My Favorite

From Frege to Gödel:

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