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A Step by Step Guide to Gödel’s Incompleteness Proof
4: The Gödel numbering system


 

 


 

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.

 

The Gödel numbering system

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, ~, , , (, and ) are matched to the numbers 1, 3, 5, 7, 9, 11 and 13 as below:

 

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 172, 192, 232, etc. In general, for a type n variable, we use 17n, 19n, 23n, etc (hideshow Gödel’s).

 

“variables of type n are given numbers of the form pn (where p is a prime number > 13).”

 

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 nth prime number to the value of the nth number in the series (hideshow Gödel’s).

 

“to every finite series of basic signs … there corresponds, one-to-one, a finite series of natural numbers. These finite series of natural numbers we now map (again in a one-to-one correspondence) on to natural numbers, by letting the number 2n1, 3n2pknk correspond to the series n1, n2, … nk, where pk denotes the k‑th prime number in order of magnitude.”

 

For example, for the sequence ~(ff0), the corresponding number series is 511331, 13. The single number corresponding to this series is 25· 311· 53· 73· 111· 1313 (where · indicates multiplication).

 

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 qn, where q is a prime number and q > 13 and n > 0, and (pq)n = pqn = (pn)q.

 

For example, (p19)17 = p19·17 = (p17)19. By the definition of Gödel numbering, this corresponds both to a type 17 variable y17 and to a type 19 variable x19.

 

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 1717, 1919, 2323, , type 3 variables of the form (1717)17, (1919)19, (2323)23, , etc. Obviously this would affect the definition of some of the functions and relations that are defined later in the proof, but in principle this is readily achievable, so this error does not affect the overall thrust of Gödel’s argument.

 

 

Correspondence of relations

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.

 

Usage of Italics

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.)

 

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

 

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 2φ[ formula1 ] · 3φ[ formula2 ] · 5φ[ formula3 ] ·… , in other words, the exponents of the prime factors of x are Gödel numbers, rather than x itself being a Gödel number. So it is important to note that the number x in this case does not correspond by Gödel numbering to a series of formulas or 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).

 

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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The Lighter Side

 

NEWS

Lebesgue Measure

There is now a new page on Lebesgue measure theory and how it is contradictory.

 

 

Illogical Assumptions

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

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).

 

 

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.

 

 

New page on Chaitin’s Constant

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  

 

13 May 2015 Good Math, Bad Math?

 

30 Apr 2015 The Chinese Room

 

31 Mar 2015 Cranks and Crackpots

 

16th Mar 2015 Bishops Dancing with Pixies?

 

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:

Gödel Links

 

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

Gödel, Minds, and Machines

 

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