Logic and Language

Logic and Language

Copyright © James R Meyer 2012 - 2017 www.jamesrmeyer.com

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6: Gödel’s Relations of Natural Numbers 1 to 23

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

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.

Note again that in Gödel’s paper the epsilon symbol **ε** is used in several definitions; the meaning of the term **ε** (see also Epsilon notation ε) is given by:

**εx R(x, y)** is a function with the free variable **y**, and whose value is *the smallest number* **x** *for which the relation* **R(x, y)** *holds*, and if there is no such smallest number, then the value of **εx R(x, y) **is **0**.

In the section entitled “*Relations 1-46*” Gödel defines various relations/functions. The relations/functions 1-45 are all primitive recursive; relation 46 is not defined as primitive recursive. Note that while in modern terminology, all variables of a relation are normally enclosed after the relation name in brackets (e.g. Rel(x, y) ), Gödel often places variables before and after the relation name (e.g. x Rel y) ). The names are mainly abbreviations of German words, see notes at the foot of this page for the words and the English translations.

Some of these functions (nos. 3, 4, 5, 16, 30) are recursively defined in terms of the value of the function itself. For example, the function 4 is the factorial function mentioned in the previous page.

For most of the relations/functions, Gödel adds comments describing the relationship between symbol strings of the formal system that corresponds (by the Gödel numbering function) to the given relation between natural numbers. In the following, further notes are included to assist the reader. In these notes variables for symbols/symbol strings of the formal system **P** are represented by colored capital letters, e.g., X, Y. (Footnote: Note that these variables are in a language that is a meta-language to the formal system.)

It is crucial to remember that all these relations/functions are purely number-theoretic - that is, that they only refer to natural numbers and variables whose domain is natural numbers. It is important not to use intuition to jump to conclusions, nor to make assumptions about the corresponding relationships between symbol strings of the formal system. Any such correspondence must follow in a logical manner.

It should be noted that in some cases, a corresponding number is obtained not by the Gödel numbering function **φ **but by the **ψ** function (as referred to previously, see The Psi function: here). This does not present any problems, since given a number obtained via the **ψ** function for a symbol, one can always obtain the Gödel number by applying the Gödel numbering function **φ** to that single symbol. For example **ψ( f) = 3**, and

**NB**: when the term ‘symbol’ is used below, unless otherwise indicated, that can mean either a single basic symbol of the system **P** or a variable of the system **P**. Also note that some of the ‘formulas’ used in the examples are not actually proper formulas of the system **P**; this is only so that the examples are not overly long.

Basic Arithmetical Concepts.

These are basic arithmetical concepts, principally concerning properties of prime numbers. Since the Gödel numbering method is based on the fact that every natural number has a unique factorization into its constituent prime factors, these relations/functions underpin the subsequent relations and functions. The details of the definitions are given below; clicking “show” will show the details for that definition.

1. **x/y** (show)

2. **Prim(x)** (show)

3. **n Pr x** (show)

4. **n!** (show)

5. **Pr(n)** (show)

Relations/functions that correspond to the construction of symbol strings

We now move from basic arithmetical definitions to the definition of functions that correspond to operations on symbols and symbol strings of the formal system - and to the definition of relations that correspond to assertions regarding symbol strings of the formal system.

The formulas of the formal system are combinations of the symbols of the system that satisfy certain conditions. One of the goals of these relations and functions is to lead to the definition of relation 23, which corresponds to the assertion that a given symbol string is a formula of the formal system.

It will be noted that the functions/relations are presented here in a different order to that given by Gödel; here they are grouped according to their similarity and purpose.

These are basic functions that correspond to:

6. **n Gl x**: the operation of obtaining the symbol at a particular position in a symbol string

7. **l(x)**: the operation of counting the number of symbols in a symbol string

9. **R(x)**: the operation of obtaining the Gödel number of a single symbol

6. **n Gl x** (show)

7. **l(x)** (show)

9. **R(x)** (show)

**x * y** corresponds to the operation of concatenating (joining together) two symbol strings.

8. **x * y** (show)

These are functions that correspond to:

16. **n N x**: the operation of **n** repetitions of prefixing a symbol string by the symbol **f**

17. **Z(n)**: the operation of **n** repetitions of prefixing the symbol **0** by the symbol **f**

16. **n N x** (show)

17. **Z(n)** (show)

These functions correspond to operations that build up more complex symbol strings from simpler ones. They correspond to:

10. **E(x)**: the operation of putting brackets around a symbol string

13. **Neg(x)**: the operation of creating the negation of a symbol string

14. **x Dis y**: the operation of joining two symbol strings by the ‘or’ symbol **∨**

15. **x Gen y**: the operation of prefixing a symbol string by a variable and the ‘for all’ symbol **∀**

10. **E(x)** (show)

13. **Neg(x)** (show)

14. **x Dis y** (show)

15. **x Gen y** (show)

Relations 11. **n Var x** and 12. **Var(x)** correspond to the assertion that a particular symbol is a variable (reminder: we refer to a variable of the formal system as a ‘single’ symbol, although it is actually composed of two or more symbols).

11. **n Var x** (show)

12. **Var(x)** (show)

Relations 18. **Typ _{1}′(x)** and 19.

18. **Typ _{1}′(x) **19.

Relations 20. **Elf(x)** and 21. **Op(x,y,z)** correspond to the assertion that a particular combination of symbols is a particular type of formula of the formal system. **FR(x)** is a relation which corresponds, though not directly by Gödel numbering, to the notion of a series of formulas. These relations are not used elsewhere; their only purpose is for the definition of **Form(x)**.

Relation 23. **Form(x) **corresponds to the assertion that a particular combination of symbols is a formula of the formal system. It is defined in terms of **Elf(x)**, **Op(x,y,z)** and **FR(x)**.

20. **Elf(x) **21. **Op(x,y,z) **22. **FR(x)** 23. **Form(x)** (show)

Footnotes:

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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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.

There is a new addition to the page Yet another flawed incompleteness proof, where Berto’s proof of incompleteness in his book *There’s something about Gödel* comes under scrutiny.

I found that making, adding or deleting footnotes in the traditional manner proved to be a major pain. So I developed a different system for footnotes which makes inserting or changing footnotes a doddle. You can check it out at Easy Footnotes for Web Pages (Accessibility friendly).

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.

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.

16th Mar 2015 Bishops Dancing with Pixies?

23rd Feb 2015 Artificial Intelligence

31 Mar 2015 Cranks and Crackpots

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:

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

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Copyright © James R Meyer 2012 - 2017

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