GÃ¶dels Incompleteness Theorems Kurt GÃ¶del His Theorems A ...

Introduction Gödel Numbering Meta-mathematics Gödels Proofs Conclusions 

Kurt Gödel 

Gödels Incompleteness Theorems 

Sebastian Bader 1 

ICCL, Technische Universität Dresden, Germany 

Knowledge Representation and Reasoning Seminar 

April 25th, 2006 

◮ Born: April 28, 1906 in 

Brno, Austria-Hungary 

(now Czech Republic) 

◮ Died: January 14, 1978 in 

Princeton, New Jersey, 

USA 

◮ 1930: PhD (completeness 

of the first-order predicate 

calculus) 

◮ 1932: ”Über formal 

unentscheidbare Sätze der 

Principia Mathematica und 

verwandter Systeme I.” 

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


His Theorems 

Gödels Incompleteness Theorems (Sebastian Bader ) 2 


A sufficient system 

“Über formal unentscheidbare Sätze 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) 


arithmetical truths, the statement of its own consistency can be 

deduced (within the system) if and only if it is inconsistent. 

A formal system capable of proving basic arithmetical truths 

(like all valid additions, multiplications, ...) must contain: 

◮ an axiomatisation of the natural numbers. 

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

functions over natural numbers built of 

• all projections π i 

• composition of functions 

• primitive recursion: let f and g be primitive recursive, then 

h(0, x 0 , ..., x k−1 ) = f (x 0 , ..., x k−1 ) and 

h(s(n), x 0 , ..., x k−1 ) = g(h(n, x 0 , ..., x k−1 ), n, x 0 , ..., x k−1 ) 

is primitive recursive. 



Gödels System P 



General Outline of the Proofs 

◮ Peano’s Aximos: ¬(sx = 0), (sx = sy) ⊃ (x = y), 

(p(0) ∧ (∀x)(p(x) ⊃ p(sx))) ⊃ ((∀x)p(x)) 

◮ Structural schemata: (p ∨ p) ⊃ p, p ⊃ (p ∨ q), 

(p ∨ q) ⊃ (q ∨ p), (p ⊃ q) ⊃ ((r ∨ p) ⊃ (r ∨ q)), 

◮ Substitution schemata: (∀v)a ⊃ a{v ↦→ c}, 

(∀v)(b ∨ a) ⊃ (b ∨ (∀v)a), 

◮ Axiom of reducibility (comprehension axiom of set theory) 

(∃p)(∀x)(p(x) ≡ a), 

◮ Axiom of extensionality (Two sets are the same if and only 

if they have the same elements) 

(∀x)(p(x) ≡ q(x)) ⊃ (p = q) 

◮ Let PM be a system, capable of proving basic 

mathematical truths. 

◮ Map symbols, formulae and proofs to natural numbers. 

◮ Map meta-mathematical statements to properties of 

(relations between) natural numbers. 

◮ Therefore, we can talk about those statements within the 

system PM. 

◮ 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. 

◮ Map symbols to numbers. 

◮ Map variables to numbers. 

◮ Map formulae to numbers. 

◮ Map proofs (sequences of formulae) to numbers. 



Map symbols to numbers 

Symbol Gödel 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 


Gödels Incompleteness Theorems (Sebastian Bader ) 8


Map variables to numbers 


Map formulae to numbers 

There are three kinds of variables: 

◮ Numerical variables (x, y, z): ranging over numerals 

0, s0, ... or numerical expressions like x + y 

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

(sentences) 

◮ Predicate variables (P, Q, R): ranging over predicates 

Type Examples Gödel numbers 

Numerical x, y, z, . . . 13, 17, 19, . . . 

Sentential p, q, r, . . . 13 2 , 17 2 , 19 2 , . . . 

Predicate P, Q, R, . . . 13 3 , 17 3 , 19 3 , . . . 

( ∃ x ) ( x = s y ) 

↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ 

8 4 13 9 8 13 5 7 17 9 

↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓ 

m = 2 8 ×3 4 ×5 13 ×7 9 ×11 8 ×13 13 ×17 5 ×19 7 ×23 17 ×29 9 

We call m the Gödel number of (∃x)(x = sy). 



From numbers to formulae 



Map proofs to numbers 

Can we go backwards? Yes if a number is a valid Gödel 

number then we can construct the corresponding formula: 

243,000,000 

↓ 

2 6 3 5 5 6 

↓ ↓ ↓ 

6 5 6 

↓ ↓ ↓ 

0 = 0 

Formula 

(∃x)(x = sy) 

(∃x)(x = s0) 

Gödel number 

m 

n 

The Gödel number of the (part of the) proof 

is k = 2 m × 3 n . 

(∃x)(x = sy) 

(∃x)(x = s0) 



Properties 



The Correspondence Lemma 

◮ For every formula in PM there is a unique Gödel number. 

◮ For every proof in PM there is a unique Gödel number. 

◮ The Gödel-number-function and its inverse are 

computable. 

Lemma 

Every primitive recursive truth, when represented as a string of 

symbols, is a theorem of PM. 



A simple typographic property 



A simple typographic property, ctd 

◮ Gödel numbers map formulae / proofs to natural numbers. 

◮ But we can also express statements about structural 

properties using numbers. 

Example 

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

Example 

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

◮ How can this be expressed using arithmetics? 

◮ The Gödel number of the formula ¬(0 = 0) is 

a = 2 1 × 3 8 × 5 6 × 7 5 × 11 6 × 13 9 . 

◮ We can state that the exponent of 2 (first position, hence 

smallest prime) in a’s prime factorisation is 1 (Gödel 

number of ¬ = 1). 

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





Example 

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

◮ 2 is a factor of a, but 2 2 is not. 

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

(∃z)(y = z × x). 

◮ Hence, we obtain the formula T : 

(∃z)(s 

} 

. 

{{ 

. . s0 

} 

= z × 

}{{} 

ss0) ∧ ¬(∃z)(s 

} 

. 

{{ 

. . s0 

} 

a 

2 

a 

= z × (ss0 

} {{ 

× ss0 

} 

)) 

2 2 

Since the predicate “x is factor of y” is primitive recursive and 

the statement is true, T is a theorem of PM. 



Example 

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

◮ Is equivalent to the theorem T : 

(∃z)(s 

} 

. 

{{ 

. . s0 

} 

= z × 

}{{} 

ss0) ∧ ¬(∃z)(s 

} 

. 

{{ 

. . s0 

} 

a 

2 

a 

= z × (ss0 

} {{ 

× ss0 

} 

)) 

2 2 

◮ We can talk about typographic properties by talking about 

properties of the prime factorization of (large) integers. 



Another typographic property 



Another typographic property 

Example 

The sequence of formulae with Gödel number x is a proof (in 

PM) of the formula with Gödel number z. 

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

◮ Yes, there is an arithmetical (but by no means simple) 

relation between x and z. 

◮ We will use dem(x, z) to denote: 

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

Some remarks: 

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

◮ dem is primitive recursive. 

◮ Therefore, for each valid dem(x, y) there is a theorem in 

PM, denoted Dem(x, y). 

◮ PM has the capability to talk accurately about itself. 



The substitution statement 



The stage 

◮ Assume that (∃x)(x = sy) has the Gödel number m. 

? What do we get if we replace y (Gödel number 17) with m? 

◮ The formula (∃x)(x = s 

} 

s . 

{{ 

. . s0 

} 

). 

m 

◮ This formula has again a Gödel number (even larger), 

denoted by sub(m, 17, m). 

◮ Again, sub is primitive recursive. 

◮ Therefore, for each a = sub(x, 17, x) there is a string 

Sub(s 

} 

. 

{{ 

. . s0 

} 

, 

} 

s . 

{{ 

. . s0 

} 

, 

} 

s . 

{{ 

. . s0 

} 

) whose evaluation is a. 

m 17 m 

◮ We can map formulae and proofs to numbers. 

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

◮ We can refer to the Gödel number of the formula obtained 

by substituting the Gödel number of some formula into 

itself using sub(m, 17, m). 



Outline of the proofs 



Step (i) 

(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 

Construct a formula G stating: “The formula G is not 

demonstrable using the rules of PM”. 

◮ Recall Dem(x, z) states: The sequence of formulae with 

Gödel number x is a proof of the formula with number z. 

◮ Therefore, (∃x) Dem(x, z) states: The formula with number 

z is demonstrable. 

◮ Therefore, ¬(∃x) Dem(x, z) states: The formula with 

number z is not demonstrable. 

◮ We will show that a special case of this formula is not 

demonstrable. 




The Formula 


The Formula, ctd. 

◮ Consider the formula 

¬(∃x) Dem(x, Sub(y, 17, y)) 

and let n be its Gödel number. (Meaning: The formula with 

Gödel number sub(y, 17, y) is not demonstrable.) 

◮ Substituting n for y we obtain: 

¬(∃x) Dem(x, Sub(n, 17, n)) 

(Meaning: The formula with Gödel number sub(n, 17, n) is 

not demonstrable.) 

(F) 

(G) 

◮ Let G be 

¬(∃x) Dem(x, Sub(n, 17, n)) 

and let g denote its Gödel number. 

◮ We have g = sub(n, 17, n). 

? Why: 

• sub(n, 17, n) is the Gödel number of the formula created 

from the formula with number n, by replacing y with n. 

• But this formula is G! 

◮ Therefore, G expresses that G is not demonstrable. 



Step (ii) 



Step (ii) – Conclusions 

Theorem 

G is demonstrable iff ¬G is demonstrable. 

Sketch of the ⇒-direction. 

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

◮ Then there is a proof (with Gödel number k) of G. 

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

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

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

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

Theorem 

G is demonstrable iff ¬G is demonstrable. 

Conclusions: 

◮ If G and ¬G are demonstrable in PM, then PM is 

inconsitent. 

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



Step (iii) 



Step (iv) 

Show that G is true. 

◮ G says: There is no demonstration of G. 

◮ On the level of numbers: There is no number x which 

bears the relationship dem with the number sub(n, 17, n). 

◮ We just showed that there is no proof (hence no number x) 

for G in (a consistent) PM. 

◮ Therefore, G must be true. 

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

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

is incomplete. 

◮ We know that G is true. 

◮ We know that G can be expressed within PM. 

◮ We know that G is not demonstrable within PM. 

◮ Therefore, PM is incomplete. 

◮ Moreover, even if we add G as a statement to PM, there is 

a formula G ′ in 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. 

◮ If PM is consistent, then there is at least one formula that is 

not demonstrable in PM. 

◮ 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) 

◮ This can be expressed in PM as: 

(∃y)¬(∃x) Dem(x, y) 

(A) 



Step (v) – Conclusions 

(∃y)¬(∃x) Dem(x, y) 

¬(∃x) Dem(x, Sub(n, 17, n)) 

◮ The formula A ⊃ G is demonstrable in PM. 

◮ Therefore, A is not demonstrable (If it were, G would be 

demonstrable, which is not). 

◮ Therefore, the consistency of PM cannot be established by 

any reasoning which can be mapped into PM. 

(A) 

(G) 




Conclusion 


Misconceptions 

Theorem (First Incompleteness Theorem) 


arithmetical truths, it is possible to construct an arithmetical 

statement that is true but not provable within the system. 

Theorem (Second Incompleteness Theorem) 


arithmetical truths, the statement of its own consistency can be 

deduced (within the system) if and only if it is inconsistent. 

◮ The theorems do not imply that every interesting axiomatic 

system is incomplete. (E.g., Euclidean geometry can be 

completely axiomatized.) 

◮ The theorems apply to those systems only which allow to 

define natural numbers. 

◮ The theorems apply to those systems only which are used 

as their own proof systems. 



Implications 



By the way ... 

◮ 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.) 

◮ Unfortunately, a proof in T is not completely convincing 

without known T to be consistent. ... 

Gödel showed some slightly weeker versions of the theorems. 

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

Thanks for your attention.

GÃ¶dels Incompleteness Theorems Kurt GÃ¶del His Theorems A ...

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