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A Step by Step Guide to Gödel’s Incompleteness Proof:

7: Relations of Natural Numbers 24 to 46

Page last updated 14 May 2021

Note that (provided you have JavaScript enabled) clicking on (show hide ) will reveal further details, while clicking again will hide it. Also, clicking on (show Gödel’s hide Gödel’s ) will reveal relevant parts of Gödel’s text (shown in green), while clicking again will hide it.

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.

### 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.

**Sb**31.

_{k}(x^{v}/_{y})**Sb(x**( hide show)

^{v}/_{y})

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.

### 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**
( hide show)

35. **A _{1}-Ax(x)**,

**A**,

_{2}-Ax(x)**A**,

_{3}-Ax(x)**A**36.

_{4}-Ax(x)**A-Ax(x)**( hide show)

### 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.

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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