Copyright   James R Meyer    2012 - 2023 https://www.jamesrmeyer.com

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

Page last updated 28 Dec 2022

Now printer friendly: this guide has now been set up for easy printing so that readers can also access the information in paper format.

Note that (provided you have JavaScript enabled) clicking on () will reveal further details, while clicking again will hide it.

(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.

## 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   ()

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 x1 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.

In Gödel’s paper…

25. v Fr n,x   ()

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 ]

In Gödel’s paper…

26. v Fr x   ()

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 ].

In Gödel’s paper…

### 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)   ()

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 · 3b · 5c … · p17 · ju · kw · ly · mz · …

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, ffff 0, then

φ[ ffff 0 ] = 23 · 33 · 53 · 73 · 111

Sb(x v/y) = 2a · 3b· 5c … · p3 · j3 · k3 · l3 · m1· qu · rw · sy · tz· …

This corresponds to the substitution of the type 1 variable x1 by the type 1 string ffff 0

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, where Gödel stated:

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 string 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 term belonging to metamathematics.”

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 string type as the variable v. Hence where Sb is used in the following functions/relations, there must be the added stipulation to that effect.

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

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

In Gödel’s 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   ()

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

In Gödel’s paper…

33. n Th x   ()

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).

In Gödel’s paper…

34. Z–Ax(x)   ()

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.

In Gödel’s paper…

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

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

In Gödel’s paper…

37. Q(z,y,v)   ()

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 ].

In Gödel’s paper…

38. L1-Ax(x)   ()

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

In Gödel’s paper…

39. L2-Ax(x)   ()

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

In Gödel’s paper…

40. R-Ax(x)   ()

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

In Gödel’s paper…

41. M-Ax(x)   ()

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.

In Gödel’s paper…

42. Ax(x)   ()

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

In Gödel’s paper…

### 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)   ()

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.

In Gödel’s paper…

44. Bw(x)   ()

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

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. By trhe reference to Fl(x y z), each formula is either an axiom or a formula given by the rules of inference of the 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 ] - the exponents of the prime factors of x are Gödel numbers, rather than x itself being a Gödel number - in this case the number x does not correspond by Gödel numbering to a series of formulas nor to a proof schema of the formal system P, i.e:

x  ≠  φ (series of formulas of the formal system P).

x  ≠  φ (proof-schema of the formal system P).

For more on this, see Corresponding Relations in Part 4 of this guide.

In Gödel’s paper…

45. x B y   ()

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

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[ Xn ]

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

In Gödel’s paper…

46. Bew(x)   ()

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

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[ Yn ]

In Gödel’s paper…

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

You can help by sharing the site with others. You can also donate at where there are full details.

As site owner I reserve the right to keep my comments sections as I deem appropriate. I do not use that right to unfairly censor valid criticism. My reasons for deleting or editing comments do not include deleting a comment because it disagrees with what is on my website. Reasons for exclusion include:
Comments with excessive number of different points.
Questions about matters that do not relate to the page they post on. Such posts are not comments.
Comments with a substantial amount of mathematical terms not properly formatted will not be published unless a file (such as doc, tex, pdf) is simultaneously emailed to me, and where the mathematical terms are correctly formatted.

Reasons for deleting comments of certain users:
Bulk posting of comments in a short space of time, often on several different pages, and which are not simply part of an ongoing discussion. Multiple anonymous user names for one person.
Users, who, when shown their point is wrong, immediately claim that they just wrote it incorrectly and rewrite it again - still erroneously, or else attack something else on my site - erroneously. After the first few instances, further posts are deleted.
Users who make persistent erroneous attacks in a scatter-gun attempt to try to find some error in what I write on this site. After the first few instances, further posts are deleted.

Difficulties in understanding the site content are usually best addressed by contacting me by e-mail. Based on HashOver Comment System by Jacob Barkdull Copyright   James R Meyer   2012 - 2023
https://www.jamesrmeyer.com