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onate create account log in personal tools donate create account log in contents move to sidebar hide top 1 simplified overview 2 gödel s encoding toggle gödel s encoding subsection 2 1 example 3 lack of uniqueness 4 application to formal arithmetic toggle application to formal arithmetic subsection 4 1 recursion 4 2 expressing statements and proofs by numbers 5 generalizations 6 gödel sets 7 see also 8 notes 9 references 10 further reading toggle the table of contents gödel numbering 20 languages العربية বাংলা català deutsch ελληνικά esperanto español français עברית italiano 日本語 한국어 nederlands português русский simple english slovenščina українська 粵語 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia function in mathematical logic for numberings of the set of computable functions see numbering computability theory this article includes a list of references related reading or external links but its sources remain unclear because it lacks inline citations please help improve this article by introducing more precise citations april 2021 learn how and when to remove this message this article needs more citations please help improve this article by adding citations to reliable sources unsourced material may be challenged and removed find sources gödel numbering news newspapers books scholar jstor may 2025 learn how and when to remove this message 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 kurt gödel developed the concept for the proof of his incompleteness theorems 1 173 198 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 edit 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 foxy is represented by 102 111 120 121 the logical formula x y y x is represented by 120 061 121 032 061 062 032 121 061 120 gödel s encoding edit number variables property variables symbol 0 s x 1 x 2 x 3 p 1 p 2 p 3 number 1 3 5 7 9 11 13 17 19 23 289 361 529 gödel s original encoding 1 179 i 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 displaystyle 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 e n c x 1 x 2 x 3 x n 2 x 1 3 x 2 5 x 3 p n x n displaystyle 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 proof s key observation gödel 1931 there are more sophisticated and more concise ways to construct a gödel numbering for sequences example edit 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 6 3 5 5 6 243 000 000 lack of uniqueness edit 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 base k numeral system a formula consisting of a string of n symbols s 1 s 2 s 3 s n displaystyle s_ 1 s_ 2 s_ 3 dots s_ n would then be mapped to the number h s 1 k n 1 h s 2 k n 2 h s n 1 k 1 h s n k 0 displaystyle 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 if k is chosen to be a power of 10 this scheme makes it fairly easy for a human to convert between a string of symbols and its gödel number since the gödel number represented in base 10 is just the concatenation of the n displaystyle n decimal numbers h s i displaystyle h s_ i for example the numbering described here has k 1000 ii application to formal arithmetic edit recursion edit 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 edit 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 r is an inference rule then there should be some arithmetical function g r of natural numbers such that if formula c is derived from formulas a and b through an inference rule r i e a b r c displaystyle a b vdash _ r c then g r f a f b f c displaystyle 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 edit 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 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 citation needed 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 citation needed gödel sets edit 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 see also edit church encoding description number gödel numbering for sequences gödel s incompleteness theorems chaitin s incompleteness theorem notes edit gödel s notation 1 176 has been adapted to modern notation 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 h s n n displaystyle h s_ n n and the formula above 1 10 3 1 2 10 3 2 3 10 3 3 1 10 2 2 10 1 3 10 0 100 20 3 displaystyle 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 iii we arrive at 123 displaystyle 123 as our numbering a neat feature or in bijective base 10 form 9 a 1 a 3 displaystyle 9a 1a 3 references edit 1 2 3 gödel kurt 1931 über formal unentscheidbare sätze der principia mathematica und verwandter systeme i pdf monatshefte für mathematik und physik in german 38 173 198 doi 10 1007 bf01700692 s2cid 197663120 archived from the original pdf on 2018 04 11 retrieved 2013 12 07 nagel ernest newman james r 1959 gödel s proof further reading edit hofstadter douglas 1979 gödel escher bach an eternal golden braid basic books isbn 978 0 465 02656 2 defines and uses an alternative gödel numbering hofstadter douglas 2007 i am a strange loop basic books isbn 978 0 465 03078 1 includes the history of gödel s numbering hemann jason holk eric june 15 2013 visualizing the turing tarpit pdf proceedings of the first acm sigplan workshop on functional art music modeling design pp 71 76 doi 10 1145 2505341 2505348 isbn 978 1 4503 2386 4 archived from the original pdf on september 27 2016 uses gödel numbering to encode programs v t e mathematical logic general axiom list cardinality first order logic formal proof formal semantics foundations of mathematics information theory lemma logical consequence model theorem theory type theory theorems list paradoxes gödel s completeness incompleteness theorems tarski s undefinability banach tarski paradox cantor s theorem paradox diagonal argument compactness halting problem lindström s löwenheim skolem russell s paradox logics traditional classical logic logical truth tautology proposition inference logical equivalence consistency equiconsistency argument soundness validity syllogism square of opposition venn diagram propositional boolean algebra boolean functions logical connectives propositional calculus propositional formula truth tables many valued logic 3 finite predicate first order list second order monadic higher order fixed point free quantifiers predicate monadic predicate calculus set theory set hereditary class ur element ordinal number extensionality forcing relation equivalence partition set operations intersection union complement cartesian product power set identities types of sets countable uncountable empty inhabited singleton finite infinite transitive ultrafilter recursive fuzzy universal universe constructible grothendieck von neumann maps cardinality function map domain codomain image in sur bi jection schröder bernstein theorem isomorphism gödel numbering enumeration large cardinal inaccessible aleph number operation binary theories zermelo fraenkel axiom of choice continuum hypothesis general kripke platek morse kelley naive new foundations tarski grothendieck von neumann bernays gödel ackermann constructive formal systems list language syntax alphabet arity automata axiom schema expression ground extension by definition conservative relation formation rule grammar formula atomic closed ground open free bound variable language metalanguage logical connective predicate functional variable propositional variable proof quantifier rank sentence atomic spectrum signature string substitution symbol function logical constant non logical variable term theory list example axiomatic systems list of true arithmetic peano second order elementary function primitive recursive robinson skolem of the real numbers tarski s axiomatization of boolean algebras canonical minimal axioms of geometry euclidean elements hilbert s tarski s non euclidean principia mathematica proof theory formal proof natural deduction logical consequence rule of inference sequent calculus theorem systems axiomatic deductive hilbert list complete theory independence from zfc proof of impossibility ordinal analysis reverse mathematics self verifying theories model theory interpretation function of models model atomic equivalence finite prime saturated spectrum submodel non standard model of non standard arithmetic diagram elementary categorical theory model complete theory satisfiability semantics of logic strength theories of truth semantic tarski s kripke s t schema transfer principle truth predicate truth value type ultraproduct validity computability theory church encoding church turing thesis computably enumerable computable function computable set decision problem decidable undecidable p np p versus np problem kolmogorov complexity lambda calculus primitive recursive function recursion recursive set turing machine type theory related abstract logic algebraic logic automated theorem proving category theory concrete abstract category category of sets history of logic history of mathematical logic timeline logicism mathematical object philosophy of mathematics supertask mathematics portal authority control databases international gnd national united states israel retrieved from https en wikipedia org w index php title gödel_numbering oldid 1343580865 categories mathematical logic theory of computation works by kurt gödel hidden categories articles with short description short description is different from wikidata articles lacking in text citations from april 2021 all articles lacking in text citations articles needing additional references from may 2025 all articles needing additional references cs1 german language sources de all articles with unsourced statements articles with unsourced statements from february 2020 this page was last 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