Logic and Language

Logic and Language

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

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4: The Gödel numbering system

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.

In his proof Gödel used a method so that any combination of symbols of the formal system can be represented as a natural number. This method has since been called the Gödel numbering system. In the paper it is given in the paragraph beginning *“The basic signs of the system P…” .*

Gödel’s numbering system means that every formula of the formal system **P** can be represented as a number; and every proof-schema of the formal system **P** can also be represented as a number.

The Gödel numbering method is *intended* (see below) to be a one-to-one function, so that every string of symbols is matched to a unique number. No two strings can have the same number, and no two numbers can match to the same string. The numbering method can be considered as a two-step process. The first step is to match every symbol of the formal system to a specific number. The basic symbols **0**, * f*,

0 |
↔ | 1 |
f |
↔ | 3 |
~ |
↔ | 5 |
∨ |
↔ | 7 |

∀ |
↔ | 9 |
( |
↔ | 11 |
) |
↔ | 13 |

where **∨** is the symbol for ‘or’, and **∀** is the symbol for ‘for all’.

For variables, the matching is done using prime numbers greater than **13**. For **type 1** variables, we use **17**, **19**, **23**, etc. For **type 2** variables, we use **17 ^{2}**,

In this way, every basic symbol and every variable of the formal system has a unique corresponding natural number.

This is the definition of a function, and although Gödel does not give this function a name, it is useful to call this function **ψ**; for example, **ψ[∀] = 9**. We will need to refer to this function later on. The function **ψ ** gives a corresponding series of natural numbers for every symbol string of the formal system.

**NB**: It is important to bear in mind that while a variable of the formal system **P** is actually a string of symbols, for the purposes of the Gödel numbering function, every variable is considered in the same way as a single symbol. So, for convenience and to avoid undue verbosity, from this point forward, when the term ‘symbol’ is used, unless otherwise indicated, that means either a single basic symbol of the system **P** or a variable of the system **P**.

The next step is to convert such a series of natural numbers into a single natural number in a way that retains all the original information. This involves the use of prime numbers: **2**, **3**, **5**, **7**, **11**, … . The method is to raise the power of the **n**^{th} prime number to the value of the **n**^{th} number in the series (show Gödel’s).

For example, for the sequence **~(ff0)**, the corresponding number series is **5**, **11**, **3**, **3**, **1**,** 13**. The single number corresponding to this series is **2 ^{5}· 3^{11}· 5^{3}· 7^{3}· 11^{1}· 13^{13}** (where

The name Gödel gives to this function is **φ(a)**, where **a** is some combination of symbols of the formal system. Today it is commonly called the *‘Gödel numbering function’*. The inverse of this function can be applied in order to retrieve the original combination of symbols.

**Note:** The perspicacious reader may have observed that Gödel’s numbering system does not in fact ensure a unique one-to-one correspondence (and does not provide a unique Gödel number) for every symbol string of the formal system **P**. This is because in the Gödel numbering system variables of type **17** or higher correspond to a prime number **p** with the exponent **q ^{n}**, where

For example, **(p ^{19})^{17} = p^{19·17} = (p^{17})^{19}**. By the definition of Gödel numbering, this corresponds

Of course, this problem could be easily rectified, for example, by letting type **1** variables be of the form **17**, **19**, **23**, **…**, type **2** variables of the form **17 ^{17}**,

In the part beginning with *“Suppose now one is given a class or relation…”, *Gödel states that given a relationship between symbol strings of the formal system, the aim is to produce a corresponding relation between the corresponding numbers given by the function **φ**. And the aim is also that if the relationship between the symbol strings applies, then the corresponding relation between the corresponding numbers also holds. In that way, the relations between the corresponding numbers mirror precisely the relationships between the symbol strings.

**NB:** The following is a crucially important detail and the failure to observe the distinction between certain words in italics and not in italics has been a source of confusion to many people.

The Gödel numbering function **φ** gives for every symbol string a corresponding natural number. In Gödel’s paper such *numbers* are referred to by * italics*, so: (Footnote: This is the same as in the original German; in van Heijenoort’s and Hirzel’s translations, SMALL CAPITALS are used for the same purpose.)

formula |
= | φ(formula) |

axiom |
= | φ(axiom) |

provable formula |
= | φ(provable formula) |

propositional formula |
= | φ(propositional formula) |

It should also be noted that Gödel when refers to *variable*s in italics, he is not referring to the Gödel numbering function **φ**, but to the **ψ** function (see above), so:

variable |
= | ψ(variable) |

NB: Gödel also refers to numbers as a ‘*series of formulas*’ or as a *‘proof-schema’*, where a *‘proof-schema’* is a special case of a ‘*series of formulas*’. These are referred to in the relations 22 and 44 which occur later in the proof. A ‘*series of formulas*’ or a *‘proof-schema’* is a number **x** with the value ** 2 ^{φ[ formula}^{1}^{ ]} · 3^{φ[ formula}^{2}^{ ]} · 5^{φ[ formula}^{3}^{ ]} ·…** , in other words, the exponents of the prime factors of

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

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

It is crucial to remember that the same word in Gödel’s proof - in plain text - and in *italics - *represents two quite separate concepts. The failure to appreciate this distinction has led many people astray in their attempt to understand the proof. It is important to always bear in mind the distinction between a statement which * actually* states:

“There exist propositional formulas **A** of the system **P** such that neither **A** nor the negation of **A** are provable by any proof-schema of the system **P**”

and a similar statement of Gödel’s that is actually referring to natural numbers* , *such as the statement:

“There exist *propositional formulae* **a** such that neither **a** nor the *negation* of **a** are *provable formulae*”.

The above statement is not a statement that refers directly to formulas of the system **P**; it is a statement about numbers which correspond to expressions of the formal system, so that the statement is actually stating:

“There exists a number **a**, where **a = φ[A]** and **A** is a formula, such that there does not exist any number **b**, where **b = φ[B]** and **B** is a proof-schema and where **B** is a proof of **A** or **~A**.”

Because this dual use of the same word can result in confusion, it will not be used in this guide, except to refer to the italicized words in Gödel’s original paper.

Footnotes:

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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.

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.

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.

13 May 2015 Good Math, Bad Math?

31 Mar 2015 Cranks and Crackpots

16th Mar 2015 Bishops Dancing with Pixies?

23rd Feb 2015 Artificial Intelligence

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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