View source for Gödel numbering

{{Short description|Function in mathematical logic}}
{{For|numberings of the set of computable functions|Numbering (computability theory)}}
{{more footnotes|date=April 2021}}

In [[mathematical logic]], a '''Gödel numbering''' is a [[function (mathematics)|function]] that assigns to each symbol and [[well-formed formula]] of some [[formal language]] a unique [[natural number]], called its '''Gödel number'''. [[Kurt Gödel]] developed the concept for the proof of his [[Gödel's incompleteness theorems|incompleteness theorems]]. ({{harvnb|Gödel|1931}})

A Gödel numbering can be interpreted as an [[Semantics encoding|encoding]] in which a number is assigned to each [[symbol]] of a [[mathematical notation]], after which a sequence of [[natural number]]s 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 each statement within a system can be represented by a natural number (its ''Gödel number''). The significance of this was that properties of a statement—such as its truth or falsehood—would be equivalent to determining whether its Gödel number had certain properties. The numbers involved might be very large indeed, but this is not a barrier; all that matters is that such numbers can be constructed.

In simple terms, Gödel devised a method by which every formula or statement that can be formulated in the system gets a unique number, in such a way that formulas and Gödel numbers can be mechanically converted back and forth. There are many ways to do this. A simple example is the way in which English is stored as a sequence of numbers in computers using [[ASCII]]. Since ASCII codes are in the range 0 to 127, it is sufficient to pad them to 3 decimal digits and then to concatenate them:
* The word '''{{mono|1=foxy}}''' is represented by {{value|102111120121}}.
* The logical formula '''{{code|1=x=y => y=x}}''' is represented by {{value|120061121032061062032121061120}}.

== Gödel's encoding ==
{| class=wikitable style="float:right;"
|-
| COLSPAN=8 |
! COLSPAN=4 | number variables
! COLSPAN=3 | property variables
| ...
|-
! Symbol
| 0
| ''s''
| ¬
| ∨
| ∀
| (
| )
| ''x''<sub>1</sub>
| ''x''<sub>2</sub>
| ''x''<sub>3</sub>
| ...
| ''P''<sub>1</sub>
| ''P''<sub>2</sub>
| ''P''<sub>3</sub>
| ...
|-
! Number
| 1
| 3
| 5
| 7
| 9
| 11
| 13
| 17
| 19
| 23
| ...
| 289
| 361
| 529
| ...
|+Gödel's original encoding<ref>See Gödel 1931, p. 179; Gödel's notation (see p. 176) has been adapted to modern notation.</ref>
|}
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 <math>(x_1,x_2,x_3,...,x_n)</math> 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:

:<math>\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}.</math>

According to the [[fundamental theorem of arithmetic]], any number (and, in particular, a number obtained in this way) can be uniquely factored into [[prime factor]]s, 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 proof's key observation ({{harvnb|Gödel|1931}}).

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 2<sup>6</sup> × 3<sup>5</sup> × 5<sup>6</sup> = 243,000,000.

== Lack of uniqueness ==

Infinitely many different Gödel numberings are possible. 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 numeration|bijective base-''K'' numeral system]]. A formula consisting of a string of ''n'' symbols <math> s_1 s_2 s_3 \dots s_n</math> would then be mapped to the number

:<math> 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 .</math>

In other words, by placing the set of ''K'' basic symbols in some fixed order, such that the <math>i</math>-th symbol corresponds uniquely to the <math>i</math>-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 [[Proof sketch for Gödel's first incompleteness theorem#G.C3.B6del numbering|here]] has K=1000.{{efn-lr|name=fn1}}

== Application to formal arithmetic ==

===Recursion===
{{Main|Course-of-values recursion}}
One may use Gödel numbering to show how functions defined by [[course-of-values recursion]] are in fact [[primitive recursive function]]s.

===Expressing statements and proofs by numbers===
{{Main|Proof sketch for Gödel's first incompleteness theorem}}
Once a Gödel numbering for a formal theory is established, each [[rule of inference|inference rule]] of the theory can be expressed as a function on the natural numbers. If ''f'' is the Gödel mapping and ''r'' is an inference rule, then there should be some [[arithmetical function]] ''g<sub>r</sub>'' of natural numbers such that if formula ''C'' is derived from formulas ''A'' and ''B'' through an inference rule ''r'', i.e.
:<math> A, B \vdash_r C, </math>
then
:<math> g_r(f(A),f(B)) = f(C). </math>
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 system]]s.

== 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.{{Citation needed|date=February 2020}}
#More generally, an assignment of elements from a countable mathematical object, such as a countable [[group (mathematics)|group]], to natural numbers to allow algorithmic manipulation of the mathematical object.{{Citation needed|date=February 2020}}

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 machine]]s that manipulate strings rather than numbers.{{Citation needed|date=February 2020}} 
<!--
There is a very different meaning of the term Gödel numbering relating to 
[[numbering (computability theory)|numberings]] of the class of [[computable function]]s. COMMENTED OUT PER TALK -->

==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 logic|infinitary languages]].

== See also ==
* [[Church encoding]]
* [[Description number]]
* [[Gödel numbering for sequences]]
* [[Gödel's incompleteness theorems]]
* [[Kolmogorov complexity#Chaitin.27s incompleteness theorem|Chaitin's incompleteness theorem]]

== Notes ==
{{notelist-lr|refs=
{{efn-lr|name=fn1|For another, perhaps-more-intuitive example, suppose you have three symbols to encode, and choose bijective base-10 for familiarity (so enumeration starts at 1, 10 is represented by a symbol e.g. ''A,'' and place-value carries at 11 rather than 10: decimal ''19'' will still be ''19,'' and so with ''21;'' but decimal ''20'' will be ''1A''). Using <math> h(s_n) = n</math> and the formula above:
{{pb}}
<math>(1 \times 10^{(3-1)} + 2 \times 10^{(3-2)} + 3 \times 10^{(3-3)}) = (1 \times 10^{2} + 2 \times 10^{1} + 3 \times 10^{0}) = (100 + 20 + 3)</math>{{efn-lr|name=fn2}}
{{pb}}
...we arrive at '''<math>123</math>''' as our numbering—a neat feature.}}
{{efn-lr|name=fn2|(or, in bijective base-10 form: <math>9A + 1A + 3</math>)}}}}

== References ==

* {{Citation|last=Gödel|first=Kurt|title=Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I|journal=[[Monatshefte für Mathematik und Physik]]|volume=38|year=1931|pages=173&ndash;198|doi=10.1007/BF01700692|s2cid=197663120|url=http://www.w-k-essler.de/pdfs/goedel.pdf|access-date=2013-12-07|archive-url=https://web.archive.org/web/20180411113347/http://www.w-k-essler.de/pdfs/goedel.pdf|archive-date=2018-04-11|url-status=dead}}.
*''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.

{{Reflist}}

== Further reading ==
* ''[[Gödel, Escher, Bach|Gödel, Escher, Bach: an Eternal Golden Braid]]'', by [[Douglas Hofstadter]]. This book defines and uses an alternative Gödel numbering.
*''[[I Am a Strange Loop]]'' by [[Douglas Hofstadter]]. This is a newer book by Hofstadter that includes the history of Gödel's numbering.
* ''[https://web.archive.org/web/2017/https://blog.theincredibleholk.org/TarpitGazer/jot-visualization.pdf Visualizing the Turing Tarpit]'' by Jason Hemann and Eric Holk. Uses Gödel numbering to encode programs.<!-- DOI and ref data can be found here: http://dl.acm.org/citation.cfm?id=2505348 -->

{{Mathematical logic}}
{{Authority control}}

{{DEFAULTSORT:Godel numbering}}
[[Category:Mathematical logic]]
[[Category:Theory of computation]]
[[Category:Works by Kurt Gödel|Numbering]]