G ¨odels Incompleteness Theorems

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G ¨odels Incompleteness Theorems

Sebastian Bader¹

ICCL, Technische Universit ¨at Dresden, Germany

Knowledge Representation and Reasoning Seminar April 25th, 2006

1Supported by the GK334 of the German Research Foundation (DFG)

Kurt G ¨odel

I Born: April 28, 1906 in Brno, Austria-Hungary (now Czech Republic)

I Died: January 14, 1978 in Princeton, New Jersey, USA

I 1930: PhD (completeness of the first-order predicate calculus)

I 1932: ” ¨Uber formal unentscheidbare S ¨atze der Principia Mathematica und verwandter Systeme I.”

G ¨odels Incompleteness Theorems (Sebastian Bader ) 2

Introduction G ¨odel Numbering Meta-mathematics G ¨odels Proofs Conclusions

His Theorems

“ ¨Uber formal unentscheidbare S ¨atze der Principia mathematica und verwandter Systeme I”, 1931:

Theorem (First Incompleteness Theorem)

For any consistent formal system capable of proving basic arithmetical truths, it is possible to construct an arithmetical statement that is true but not provable within the system.

Theorem (Second Incompleteness Theorem) For any consistent formal system capable of proving basic arithmetical truths, the statement of its own consistency can be deduced (within the system) if and only if it is inconsistent.

A sufficient system

A formal system capable of proving basic arithmetical truths (like all valid additions, multiplications, ...) must contain:

I an axiomatisation of the natural numbers.

I an axiomatisation of primitive recursive functions, i.e.

functions over natural numbers built of

• all projectionsπi

• composition of functions

• primitive recursion: let f and g be primitive recursive, then h(0,x0, ...,xk−1) =f(x0, ...,xk−1)and

h(s(n),x0, ...,xk−1) =g(h(n,x0, ...,xk−1),n,x0, ...,xk−1) is primitive recursive.

G ¨odels System P

I Peano’s Aximos:¬(sx=0),(sx=sy) ⊃ (x=y), (p(0) ∧ (∀x)(p(x) ⊃p(sx))) ⊃ ((∀x)p(x))

I Structural schemata:(p∨p) ⊃p, p⊃ (p∨q), (p∨q) ⊃ (q∨p),(p⊃q) ⊃ ((r∨p) ⊃ (r∨q)),

I Substitution schemata:(∀v)a⊃a{v7→c}, (∀v)(b∨a) ⊃ (b∨ (∀v)a),

I Axiom of reducibility (comprehension axiom of set theory) (∃p)(∀x)(p(x) ≡a),

I Axiom of extensionality (Two sets are the same if and only if they have the same elements)

(∀x)(p(x) ≡q(x)) ⊃ (p=q)

General Outline of the Proofs

I Let PM be a system, capable of proving basic mathematical truths.

I Map symbols, formulae and proofs to natural numbers.

I Map meta-mathematical statements to properties of (relations between) natural numbers.

I Therefore, we can talk about those statements within the system PM.

I Construct statements showing the incompleteness and map them into the system.

Idea

If we can reason over numbers, we might reason over arbitrary statements, once those are mapped to numbers.

I Map symbols to numbers.

I Map variables to numbers.

I Map formulae to numbers.

I Map proofs (sequences of formulae) to numbers.

Map symbols to numbers

Symbol G ¨odel number Intended meaning

¬ 1 not

∨ 2 or

⊃ 3 if ... then ...

∃ 4 there is an ...

= 5 equals

0 6 zero

s 7 immediate successor

( 8 punctuation mark

) 9 punctuation mark

, 10 punctuation mark

+ 11 plus

× 12 times

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Map variables to numbers

There are three kinds of variables:

I Numerical variables (x,y,z): ranging over numerals 0,s0, ...or numerical expressions like x+y

I Sentential variables (p,q,r ): ranging over closed formulae (sentences)

I Predicate variables (P,Q,R): ranging over predicates Type Examples G ¨odel numbers Numerical x,y,z, . . . 13,17,19, . . . Sentential p,q,r, . . . 13²,17²,19², . . . Predicate P,Q,R, . . . 13³,17³,19³, . . .

Map formulae to numbers

( ∃ x ) ( x = s y )

↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓

8 4 13 9 8 13 5 7 17 9

↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓

m= 2⁸×3⁴×5¹³×7⁹×11⁸×13¹³×17⁵×19⁷×23¹⁷×29⁹ We call m the G ¨odel number of(∃x)(x=sy).

From numbers to formulae

Can we go backwards? Yes if a number is a valid G ¨odel number then we can construct the corresponding formula:

243,000,000

↓ 2⁶ 3⁵ 5⁶

↓ ↓ ↓

6 5 6

↓ ↓ ↓

0 = 0

Map proofs to numbers

Formula G ¨odel number

(∃x)(x=sy) m

(∃x)(x=s0) n

The G ¨odel number of the (part of the) proof (∃x)(x=sy) (∃x)(x=s0) is k=2^m×3ⁿ.

Properties

I For every formula in PM there is a unique G ¨odel number.

I For every proof in PM there is a unique G ¨odel number.

I The G ¨odel-number-function and its inverse are computable.

The Correspondence Lemma

Lemma

Every primitive recursive truth, when represented as a string of symbols, is a theorem of PM.

A simple typographic property

I G ¨odel numbers map formulae / proofs to natural numbers.

I But we can also express statements about structural properties using numbers.

Example

The first symbol of the formula “¬(0=0)” is “¬”.

A simple typographic property, ctd

Example

The first symbol of the formula “¬(0=0)” is “¬”.

I How can this be expressed using arithmetics?

I The G ¨odel number of the formula¬(0=0)is a=2¹×3⁸×5⁶×7⁵×11⁶×13⁹.

I We can state that the exponent of 2 (first position, hence smallest prime) in a’s prime factorisation is 1 (G ¨odel number of¬ =1).

I In other words: 2 is a factor of a, but 2²is not.

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A simple typographic property, ctd

Example

The first symbol of the formula “¬(0=0)” (with G.n. a) is “¬”.

I 2 is a factor of a, but 2²is not.

I We can express “x is factor of y ” in PM using:

(∃z)(y=z×x).

I Hence, we obtain the formula T : (∃z)(s. . .s0

| {z }

a

=z×ss0

|{z}

2

) ∧ ¬(∃z)(s. . .s0

| {z }

a

=z× (ss0×ss0

| {z }

2²

))

Since the predicate “x is factor of y” is primitive recursive and the statement is true, T is a theorem of PM.

A simple typographic property, ctd

Example

The first symbol of the formula “¬(0=0)” (with G.n. a) is “¬”.

I Is equivalent to the theorem T : (∃z)(s. . .s0

| {z }

a

=z×ss0

|{z}

2

) ∧ ¬(∃z)(s. . .s0

| {z }

a

=z× (ss0×ss0

| {z }

2²

))

I We can talk about typographic properties by talking about properties of the prime factorization of (large) integers.

Another typographic property

Example

The sequence of formulae with G ¨odel number x is a proof (in PM) of the formula with G ¨odel number z.

? Can this be expressed using properties of x and z?

I Yes, there is an arithmetical (but by no means simple) relation between x and z.

I We will use dem(x,z)to denote:

“x is a demonstration (formal proof in PM) for z”.

Another typographic property

Some remarks:

I dem depends implicitely on all axioms and rules in PM.

I dem is primitive recursive.

I Therefore, for each valid dem(x,y)there is a theorem in PM, denoted Dem(x,y).

I PM has the capability to talk accurately about itself.

The substitution statement

I Assume that(∃x)(x=sy)has the G ¨odel number m.

? What do we get if we replace y (G ¨odel number 17) with m?

I The formula(∃x)(x=s s. . .s0

| {z }

m

).

I This formula has again a G ¨odel number (even larger), denoted by sub(m,17,m).

I Again, sub is primitive recursive.

I Therefore, for each a=sub(x,17,x)there is a string Sub(s. . .s0

| {z }

m

,s. . .s0

| {z }

17

,s. . .s0

| {z }

m

)whose evaluation is a.

The stage

I We can map formulae and proofs to numbers.

I We can express ... is a proof for ... using dem(x,z).

I We can refer to the G ¨odel number of the formula obtained by substituting the G ¨odel number of some formula into itself using sub(m,17,m).

Outline of the proofs

(i) Construct a formula G stating: “The formula G is not demonstrable using the rules of PM”.

(ii) Show that G is demonstrable iff its negation¬G is demonstrable.

(iii) Show that G is true.

(iv) Realize that G is true but not demonstrable in PM.

Therefore, PM is incomplete.

(v) Construct a formula A representing “PM is consistent” and draw some conclusions

Step (i)

Construct a formula G stating: “The formula G is not demonstrable using the rules of PM”.

I Recall Dem(x,z)states: The sequence of formulae with G ¨odel number x is a proof of the formula with number z.

I Therefore,(∃x)Dem(x,z)states: The formula with number z is demonstrable.

I Therefore,¬(∃x)Dem(x,z)states: The formula with number z is not demonstrable.

I We will show that a special case of this formula is not demonstrable.

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The Formula

I Consider the formula

¬(∃x)Dem(x,Sub(y,17,y)) (F) and let n be its G ¨odel number. (Meaning: The formula with G ¨odel number sub(y,17,y)is not demonstrable.)

I Substituting n for y we obtain:

¬(∃x)Dem(x,Sub(n,17,n)) (G) (Meaning: The formula with G ¨odel number sub(n,17,n)is not demonstrable.)

The Formula, ctd.

I Let G be

¬(∃x)Dem(x,Sub(n,17,n)) and let g denote its G ¨odel number.

I We have g=sub(n,17,n).

? Why:

• sub(n,17,n)is the G ¨odel number of the formula created from the formula with number n, by replacing y with n.

• But this formula is G!

I Therefore, G expresses that G is not demonstrable.

Step (ii)

Theorem

G is demonstrable iff¬G is demonstrable.

Sketch of the⇒-direction.

I Assume G= ¬(∃x)Dem(x,Sub(n,17,n))is demonstrable.

I Then there is a proof (with G ¨odel number k ) of G.

I Therefore, dem(k,sub(n,17,n))must be true.

I Therefore, Dem(k,Sub(n,17,n))must be a theorem.

I In PM there is a rule P(k); (∃x)P(x).

I Hence(∃x)Dem(x,Sub(n,17,n)) = ¬G is demonstrable.

Step (ii) – Conclusions

Theorem

G is demonstrable iff¬G is demonstrable.

Conclusions:

I If G and¬G are demonstrable in PM, then PM is inconsitent.

I If PM is consistent, then neither G nor¬G is demonstrable.

Step (iii)

Show that G is true.

I G says: There is no demonstration of G.

I On the level of numbers: There is no number x which bears the relationship dem with the number sub(n,17,n).

I We just showed that there is no proof (hence no number x ) for G in (a consistent) PM.

I Therefore, G must be true.

I Note: this is not a proof within PM, but a meta argument.

Step (iv)

Realize that G is true but not demonstrable in PM, therefore PM is incomplete.

I We know that G is true.

I We know that G can be expressed within PM.

I We know that G is not demonstrable within PM.

I Therefore, PM is incomplete.

I Moreover, even if we add G as a statement to PM, there is a formula G⁰in the resulting system PM’, ...

(Note, that e.g. the dem-relation would be different in PM’)

Step (v)

Construct a formula A representing “PM is consistent” and draw some conclusions.

I If PM is consistent, then there is at least one formula that is not demonstrable in PM.

I In numbers: There is at least one number y such that there is no number x bearing the relation dem to y :

(∃y)¬(∃x)dem(x,y)

I This can be expressed in PM as:

(∃y)¬(∃x)Dem(x,y) (A)

Step (v) – Conclusions

(∃y)¬(∃x)Dem(x,y) (A)

¬(∃x)Dem(x,Sub(n,17,n)) (G)

I The formula A⊃G is demonstrable in PM.

I Therefore, A is not demonstrable (If it were, G would be demonstrable, which is not).

I Therefore, the consistency of PM cannot be established by any reasoning which can be mapped into PM.

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Conclusion

Theorem (First Incompleteness Theorem)

For any consistent formal system capable of proving basic arithmetical truths, it is possible to construct an arithmetical statement that is true but not provable within the system.

Theorem (Second Incompleteness Theorem) For any consistent formal system capable of proving basic arithmetical truths, the statement of its own consistency can be deduced (within the system) if and only if it is inconsistent.

Misconceptions

I The theorems do not imply that every interesting axiomatic system is incomplete. (E.g., Euclidean geometry can be completely axiomatized.)

I The theorems apply to those systems only which allow to define natural numbers.

I The theorems apply to those systems only which are used as their own proof systems.

Implications

I To prove the consistency of a system S, one needs a more powerful system T . (E.g., the consistency of Peano’s axioms can be shown using set theory, but not using the theory of natural numbers.)

I Unfortunately, a proof in T is not completely convincing without known T to be consistent. ...

By the way ...

G ¨odel showed some slightly weeker versions of the theorems.

The versions presented are due to J. Barkley Rosser, 1936.

Thanks for your attention.