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A Step by Step Guide to Gödel’s Incompleteness Proof:
7: Relations of Natural Numbers 24 to 46


 

 


 

Note that (provided you have JavaScript enabled) clicking on (hideshow) will reveal further details, while clicking again will hide it. Also, clicking on (hideshow Gödel’s) will reveal relevant parts of Gödel’s text (shown in green), while clicking again will hide it. Please note that older browsers may not display some symbols correctly.

 

(like this)

 

(like this)

 

This guide is intended to assist in attaining a full understanding of Gödel’s proof. If there is any difficulty in following any part of the proof, please contact me and I will try to help. And if you have any suggestions as to how this guide might be improved, please contact me. This guide is intended to be read alongside the English translation of Gödel’s original proof which can be viewed online at English translation of Gödel’s original proof or as a PDF file at English translation of Gödel’s original proof, PDF file.

 

Gödel’s Relations 24 - 46 of Natural Numbers

The number-theoretic relations are now becoming more complex as they correspond to more complex statements about formulas of the formal system P. It is not intended to cover every detail of these relations, but rather to concentrate on the main points and the thrust of the argument. It would be easy to get bogged down in such details and then fail to see the wood for the trees. As noted on the previous page, the names of the relations are mainly abbreviations of German words, see notes at the foot of this page for the words and the English translations.

 

Relations 24-26: Assertions regarding variables of the formal system

These define number-theoretic relations that correspond to assertions as to whether a symbol is a free variable or a bound variable within a given symbol string.

 

24. v Geb n,x         (hideshow)

This is an assertion regarding the natural numbers v, n and x, and it corresponds to the assertion:

at the nth symbol in the formula X, if V is a variable, and if it were to be at that position, it would be a bound variable

provided that v = ψ[V] and x = φ[X].

 

Suppose we have a formula of the form X = A (x1∀ B) C, where A, B and C are symbol strings of the formal system, and where x1 is a variable, and is the quantifier symbol for ‘for all’. Then the quantifier on the variable x1 applies everywhere in the string B, but that quantifier does not apply to the string C. So the relation is asserting that the quantifier applies throughout the string B, regardless of where the variable V might be in that string.

 

v Geb n,x asserts that if v = ψ[V], where V is a variable of the formal system, and x = φ[X], where X is a formula, then if the nth prime number in x is one of p, q, r, s, , then the corresponding variable V is bound (by the quantifier ) at the corresponding positions in the formula X (hideshow Gödel’s).

 

“The variable v is bound at the nth place in x.”

 

25. v Fr n,x         (hideshow)

This is a relation. v Fr n,x is an assertion regarding the natural numbers v, n and x, and

v Fr n,x corresponds to the assertion:

V is a variable, and it occurs as the nth symbol in the formula X, and it is free at that position

provided that v = ψ[V] and x = φ[X] (hideshow Gödel’s)

 

“The variable v is free at the nth place in x.”

 

26. v Fr x         (hideshow)

This is a relation. v Fr x is an assertion regarding the natural numbers v and x, and

v Fr x corresponds to the assertion:

V is a variable, and it occurs as a free variable in the formula X

provided that v = ψ[V] and x = φ[X] (hideshow Gödel’s).

 

v occurs in x as a free variable.”

 

Functions 27-31: Defining numbers that correspond to substitution in the formal system

The functions 27-30 lead up the function 31 which corresponds to the concept of the substitution of a free variable by a symbol or symbol string of the formal system.

 

27. Su x(n|y)   28. k St v,x   29. A(v,x)   30. Sbk(x v|y)   31. Sb(x v|y)         (hideshow)

Sb(x v|y) is a function which is defined in terms of Su x(n|y), k St v,x, A(v,x) and Sbk(x v|y); these functions are not used anywhere else.

 

if x = φ[X], v = ψ[V] and y = φ[Y], then Sb(x v|y) corresponds to:

the operation of substituting the symbol V where it occurs as a free variable within the symbol string X, by the symbol string Y.

 

As an example, suppose that we have a formula X with only one free variable x1 at only one position in the formula. Then the Gödel number corresponding to that formula X will be like this :

φ[X] = 2a · 3· 5c … · p17 · ju · kw · l· m· …

where p, j, k, l, m, q, r, s and t are all prime numbers in order of size, and the values of the exponents a, b, c, , 17, u, w, y, z, are given by the function ψ on the individual symbols/variables of the formula X, where 17 = ψ[x1].

 

If the symbol string which will be substituted is say, ffff0, then

φ[ffff0] = 23 · 3· 53 · 73 · 111

Sb(x v|y) = 2a · 3b · 5c …    · p3 · j3 · k3 · l· m1 · qu · rw · s· tz · …

This corresponds to the substitution of the type 1 variable x1 by the type 1 sign ffff0

 

Gödel states that the function Sb(x v|y) corresponds to the concept Subst a(v|b), as defined in the definition of the system P. Note that any Gödel number may be substituted for the variable y, so that the function Sb corresponds to the substitution of a free variable by any symbol string of the formal system. Subst defines that b must be of the same sign as the variable v. Hence where Sb is used in the following functions/relations, there must be the added stipulation to that effect (hideshow Gödel’s).

 

Sb(x v|y) is the concept Subst a(v|b)

 

“By Subst a(v|b) (where a stands for a formula, v a variable and b a sign of the same type as v) we understand the formula derived from a, when we replace v in it, wherever it is free, by b. Where v does not occur in a as a free variable, we must put Subst a(v|b) = a. Note that ‘Subst’ is a sign belonging to metamathematics.”

 

Note that, depending on what version of the translation you are using, Sb may be represented in this format:

Sb

which is the format using in Gödel’s original paper.

 

Relations/functions 32-42:
Assertions as to which numbers correspond to the axioms of the formal system

Functions 32 and 33 define some axioms of the formal system. Relations 34-42 inclusive are relations that use the previously defined relations/functions to define which Gödel numbers correspond to the axioms of the formal system.

 

32. x Imp y, x Con y, x Aeq y, v Ex y          (hideshow)

These define the logical equivalent of ‘implies’, ‘and’, ‘equivalence’, and ‘there exists’ (see also the axioms of the system P)

 

33. n Th x           (hideshow)

Given that x is a Gödel number corresponding to a formula, then n Th x is a function that gives the Gödel number of the formula that is the nth type-lift of the formula x (see Type-lift).

 

34. Z–Ax(x)         (hideshow)

These relations assert that x is the Gödel number of one of the Axioms I.1-3. Gödel asserts that there are numbers that correspond by Gödel numbering to each of these three axioms; he does this rather than give in detail how these numbers could be defined, but clearly these numbers could be calculated from the axioms by obtaining the equivalent formulation in the symbols of the formal system, and then applying the Gödel numbering function (hideshow Gödel’s).

 

“To the axioms I, 1 to 3, there correspond three determinate numbers, which we denote by z1, z2, z3.”

 

35. A1-Ax(x), A2-Ax(x), A3-Ax(x), A4-Ax(x)    36. A-Ax(x)           (hideshow)

Relation 35, A1-Ax(x) defines that x is a Gödel number that corresponds to an axiom defined by Axiom Schema II.1. Similarly for A2-Ax(x), A3-Ax(x) and A4-Ax(x) for axioms defined by the Axiom Schemas II.2-4.

 

Relation 36, A-Ax(x) defines that x is a Gödel number that corresponds to one of the axioms given by the Axiom Schemas II.1-4

 

37. Q(z,y,v)          (hideshow)

Q(z,y,v) ≡ ~(∃n,m,w)[n ≤ l(y) ∧ m ≤ l(z) ∧ w ≤ z ∧ w = m Gl z ∧ w Geb n,y ∧ v Fr n,y]

 

This relation is only used in the definition of relation 38 and not elsewhere. It corresponds to the assertion:

the symbol string Z does not have any variable bound at any position which is not under the influence of a quantifier on the variable V.”

provided that z = φ[Z], y = φ[Y], v = ψ[V] (hideshow Gödel’s).

 

“z contains no variable bound in y at a position where v is free.”

 

38. L1-Ax(x)          (hideshow)

This relation asserts that X is an axiom given by the Axiom Schema III.1, where x = φ[X].

 

39. L2-Ax(x)          (hideshow)

This relation asserts that X is an axiom given by the Axiom Schema III.1, where x = φ[X].

 

40. R-Ax(x)          (hideshow)

This relation asserts that X is an axiom given by the Axiom Schema IV.1, where x = φ[X].

 

41. M-Ax(x)         (hideshow)

This relation asserts that x is a number that corresponds by Gödel numbering to either the base axiom of the Axiom Schema V.1, or to a type-lift of the base axiom. As for relation 34, Z–Ax(x), Gödel asserts that there is a number that corresponds to the base axiom of Axiom Schema V.1, rather than defining it in detail; this number (and the numbers for type-lifts) could be defined from the axiom by obtaining the equivalent formulation in the symbols of the formal system, and then applying the Gödel numbering function. (hideshow Gödel’s)

 

“To the axiom V, 1 there corresponds a determinate number z4.”

 

42. Ax(x)         (hideshow)

This relation asserts that x is a number that corresponds by Gödel numbering to an axiom of the formal system P.

 

Relations 43-46:
Proofs in the formal system

The relations 43-46 deal with defining the number-theoretic relations that correspond to the concepts of the rules of inference of the system, the concept of a proof-schema, and the concept of a formula being provable in the system.

 

43. Fl(x y z)         (hideshow)

This is an assertion regarding the natural numbers x, y and z, and it corresponds to the assertion:

the formula X is derived by the rules of inference from the formulas Y and Z

where x = φ[X], y = φ[Y], z = φ[Z] and X, Y and Z are symbol strings of the formal system P (hideshow Gödel’s).

 

x is an immediate consequence of y and z

 

44. Bw(x)         (hideshow)

This relation is defined in terms of the previous relation Fl(x y z); it is an assertion that (hideshow Gödel’s)

x =  2φ[ X1 ] · 3φ[ X2 ] · 5φ[ X3 ] · …   pφ[ Xn ] .

x corresponds to a series of formulas, in that the prime factors of x have exponents that are the Gödel numbers of formulas. Each formula is either an axiom or a formula given by the rules of inference of th system applied to axioms or proven formulas.

 

NB: As for relation 22, this is not a direct correspondence by Gödel numbering and there is no symbol string of the formal system P where x = φ[X].

 

x is a proof-schema (a finite series of formulae, of which each is either an axiom or an immediate consequence of two previous ones)”

 

45. x B y         (hideshow)

This is an assertion regarding the natural numbers x and y and it corresponds to the assertion that (hideshow Gödel’s)

the symbol string Y is a formula of the formal system P and there is a proof-schema for Y that corresponds to the number x by an appropriate relation.

where y = φ[Y].

y corresponds to the exponent of the largest prime p in the number x, where:

 

x =  2φ[ X1 ] · 3φ[ X2 ] · 5φ[ X3 ] · …   pφ[ Y ]

Note that, as for relation 44 above, there is no symbol string of the formal system P where x = φ[X].

 

x is a proof of the formula y

 

46. Bew(x)         (hideshow)

This is an assertion regarding the natural numbers x and y and it corresponds to the assertion (hideshow Gödel’s):

X is a provable formula of the system P

i.e., there exists a proof-schema (a series of formulas) that is a proof of the formula X

where x = φ[X], and there exists a number w such that x is the exponent of the largest prime p in the number w, where:

 

w =  2φ[ Y1 ] · 3φ[ Y2 ] · 5φ[ Y3 ] · …   pφ[ X ]

 

x is a provable formula. [Bew(x) is the only one of the concepts 1-46 of which it cannot be asserted that it is recursive.]”

 

 

Below is a list of names used for various relations in the text, which are mostly abbreviations of German words; translations are provided below:

 

A Anzahl  = number
Aeq Aequivalenz  = equivalence
Ax Axiom  = axiom
B Beweis  = proof
Bew Beweisbar  = provable
Bw Beweisfigur  = proof-schema
Con Conjunktion  = conjunction
Dis Disjunktion  = disjunction
E Einklammern  = include in brackets
Elf Elementarformel  = elementary formula
Ex Existenz  = existence
Fl unmittelbare Folge  = immediate consequence
Flg Folgerungsmenge  = set of consequences
Form Formel  = formula
Fr frei  = free
FR Reihe von Formeln  = series of formulae
Geb gebunden  = bound
Gen Generalisation  = generalization
Gl Glied  = term
Imp Implikation  = implication
l Lange  = length
Neg Negation  = negation
Op Operation  = operation
Pr Primzahl  = prime number
Prim Primzahl  = prime number
R Zahlenreihe  = number series
Sb Substitution  = substitution
St Stelle  = place
Su Substitution  = substitution
Th Typenerhohung  = type-lift
Typ Typ  = type
Var Variable  = variable
Wid Widerspruchsfreiheit  = consistency
Z Zahlzeichen  = number-symbol

 


 

 


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The Lighter Side

 

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There is now a new page Halbach and Zhang’s Yablo without Gödel which demonstrates the illogical assumptions used by Halbach and Zhang.

 

 

Peter Smith’s ‘Proof’

It has come to my notice that, when asked about the demonstration of the flaw in his proof (see A Fundamental Flaw in an Incompleteness Proof by Peter Smith PDF), Smith refuses to engage in any logical discussion, and instead attempts to deflect attention away from any such discussion. If any other reader has tried to engage with Smith regarding my demonstration of the flaw, I would be interested to know what the outcome was.

 

 

Easy Footnotes

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O’Connor’s “computer checked” proof

I have now added a new section to my paper on Russell O’Connor’s claim of a computer verified incompleteness proof. This shows that the flaw in the proof arises from a reliance on definitions that include unacceptable assumptions - assumptions that are not actually checked by the computer code. See also the new page Representability.

 

 

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There is now a new page on Chaitin’s Constant (Chaitin’s Omega), which demonstrates that Chaitin has failed to prove that it is actually algorithmically irreducible.

 

Previous Blog Posts  

 

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23rd Feb 2015 Artificial Intelligence

 

Links  

 

For convenience, there are now two pages on this site with links to various material relating to Gödel and the Incompleteness Theorem

 

– a page with general links:

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– and a page relating specifically to the Gödel mind-machine debate:

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