Gödel numbering

For numberings of the set of computable functions, see Numbering (computability theory).

In mathematical logic, a Gödel numbering is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number, called its Gödel number. The concept was used by Kurt Gödel for the proof of his incompleteness theorems. (Gödel 1931)

A Gödel numbering can be interpreted as an encoding in which a number is assigned to each symbol of a mathematical notation, after which a sequence of natural numbers can then represent a sequence of symbols. These sequences of natural numbers can again be represented by single natural numbers, facilitating their manipulation in formal theories of arithmetic.

Since the publishing of Gödel's paper in 1931, the term "Gödel numbering" or "Gödel code" has been used to refer to more general assignments of natural numbers to mathematical objects.

Simplified overview

Gödel noted that statements within a system can be represented by natural numbers. The significance of this was that properties of statements - such as their truth and falsehood - would be equivalent to determining whether their Gödel numbers had certain properties. The numbers involved might be very long indeed (in terms of number of digits), but this is not a barrier; all that matters is that we can show such numbers can be constructed.

In simple terms, we devise a method by which every formula or statement that can be formulated in our system gets a unique number, in such a way that we can mechanically convert back and forth between formulas and Gödel numbers. Clearly there are many ways this can be done. Given any statement, the number it is converted to is known as its Gödel number. A simple example is the way in which English is stored as a sequence of numbers in computers using ASCII or Unicode:

The word HELLO is represented by 72-69-76-76-79 using decimal ASCII.
The logical statement x=y => y=x is represented by 120-61-121-32-61-62-32-121-61-120 using decimal ASCII.

Gödel's encoding

Gödel used a system based on prime factorization. He first assigned a unique natural number to each basic symbol in the formal language of arithmetic with which he was dealing.

To encode an entire formula, which is a sequence of symbols, Gödel used the following system. Given a sequence $(x_1,x_2,x_3,...,x_n)$ of positive integers, the Gödel encoding of the sequence is the product of the first n primes raised to their corresponding values in the sequence:

\mathrm{enc}(x_1,x_2,x_3,\dots,x_n) = 2^{x_1}\cdot 3^{x_2}\cdot 5^{x_3}\cdots p_n^{x_n}.\,

According to the fundamental theorem of arithmetic, any number (and, in particular, a number obtained in this way) can be uniquely factored into prime factors, so it is possible to recover the original sequence from its Gödel number (for any given number n of symbols to be encoded).

Gödel specifically used this scheme at two levels: first, to encode sequences of symbols representing formulas, and second, to encode sequences of formulas representing proofs. This allowed him to show a correspondence between statements about natural numbers and statements about the provability of theorems about natural numbers, the key observation of the proof.

There are more sophisticated (and more concise) ways to construct a Gödel numbering for sequences.

Example

In the specific Gödel numbering used by Nagel and Newman, the Gödel number for the symbol "0" is 6 and the Gödel number for the symbol "=" is 5. Thus, in their system, the Gödel number of the formula "0 = 0" is 2⁶ × 3⁵ × 5⁶ = 243,000,000.

Lack of uniqueness

A Gödel numbering is not unique, in that for any proof using Gödel numbers, there are infinitely many ways in which these numbers could be defined.

For example, supposing there are K basic symbols, an alternative Gödel numbering could be constructed by invertibly mapping this set of symbols (through, say, an invertible function h) to the set of digits of a bijective base-K numeral system. A formula consisting of a string of n symbols $s_1 s_2 s_3 \dots s_n$ would then be mapped to the number

h(s_1) \times K^{(n-1)} + h(s_2) \times K^{(n-2)} + \cdots + h(s_{n-1}) \times K^1 + h(s_n) \times K^0 .

In other words, by placing the set of K basic symbols in some fixed order, such that the i^th symbol corresponds uniquely to the i^th digit of a bijective base-K numeral system, each formula may serve just as the very numeral of its own Gödel number.

For example, the numbering described here has K=10000.

Application to formal arithmetic

Recursion

Main article: Course-of-values recursion

One may use Gödel numbering to show how functions defined by course-of-values recursion are in fact primitive recursive functions.

Expressing statements and proofs by numbers

Main article: Proof sketch for Gödel's first incompleteness theorem

Once a Gödel numbering for a formal theory is established, each inference rule of the theory can be expressed as a function on the natural numbers. If f is the Gödel mapping and if formula C can be derived from formulas A and B through an inference rule r; i.e.

A, B \vdash_r C, \

then there should be some arithmetical function g_r of natural numbers such that

g_r(f(A),f(B)) = f(C). \

This is true for the numbering Gödel used, and for any other numbering where the encoded formula can be arithmetically recovered from its Gödel number.

Thus, in a formal theory such as Peano arithmetic in which one can make statements about numbers and their arithmetical relationships to each other, one can use a Gödel numbering to indirectly make statements about the theory itself. This technique allowed Gödel to prove results about the consistency and completeness properties of formal systems.

Generalizations

In computability theory, the term "Gödel numbering" is used in settings more general than the one described above. It can refer to:

Any assignment of the elements of a formal language to natural numbers in such a way that the numbers can be manipulated by an algorithm to simulate manipulation of elements of the formal language.
More generally, an assignment of elements from a countable mathematical object, such as a countable group, to natural numbers to allow algorithmic manipulation of the mathematical object.

Also, the term Gödel numbering is sometimes used when the assigned "numbers" are actually strings, which is necessary when considering models of computation such as Turing machines that manipulate strings rather than numbers.

Gödel sets

Gödel sets are sometimes used in set theory to encode formulas, and are similar to Gödel numbers, except that one uses sets rather than numbers to do the encoding. In simple cases when one uses a hereditarily finite set to encode formulas this is essentially equivalent to the use of Gödel numbers, but somewhat easier to define because the tree structure of formulas can be modeled by the tree structure of sets. Gödel sets can also be used to encode formulas in infinitary languages.

References

Gödel, Kurt (1931), "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I" (PDF), Monatsheft für Math. und Physik, 38: 173–198 .
Gödel's Proof by Ernest Nagel and James R. Newman (1959). This book provides a good introduction and summary of the proof, with a large section dedicated to Gödel's numbering.