Linked from

The 24 pages that link to Gödel numbering, each with the reason it gives.

• Turing machineRelated: Encoding machine descriptions as strings or numbers enables universal simulation.

• Gödel's incompleteness theoremsRelated: Turns statements about arithmetic into arithmetic, making the system able to talk about itself.

• Natural numberBroader topic: It lets arithmetic encode formal expressions and reason about them numerically.

• Halting problemRelated: Encoding programs lets a computation take another program's description as input.

• Kurt GödelBroader topic: It lets arithmetic represent the syntax and proof relations of a formal system.

• Peano arithmeticRelated: It lets arithmetic represent statements and derivations about its own formal language.

• Rice's theoremRelated: Program descriptions must be encoded so an algorithm can take them as input.

• Universal Turing machineRelated: Number encodings illustrate how finite machine descriptions can be treated as manipulable data.

• DiagonalizationRelated: Coding syntax as numbers makes diagonal self-reference possible in incompleteness proofs.

• Diagonal argumentRelated: Encoding statements numerically lets diagonal constructions refer to statements about their own codes.

• Gödel's completeness theoremRelated: It makes formal syntax and proof manipulation precise, though it is not required by the theorem's statement.

• Consistency proofRelated: Gödel used numerical codes to make a system’s proofs part of arithmetic.

• Diagonal lemmaRelated: It turns formulas into numbers that arithmetic formulas can mention.

• Self-referenceRelated: Encoding formulas as numbers lets arithmetic represent statements about its own syntax.

• Gödel sentenceRelated: It lets a sentence describe its own proof status using arithmetic.

• Tarski's undefinability theoremRelated: It lets arithmetic talk about sentences and predicates in the language whose truth is at issue.

• Kleene's recursion theoremRelated: Program descriptions must be encoded as numbers before an index transformation can act on them.

• CodeRelated: It exemplifies how syntactic objects can be represented by numbers.

• Computably enumerable setRelated: Encoding objects as numbers lets enumeration and computation apply to them.

• Löb's theoremRelated: Arithmetic encoding makes statements about proofs expressible inside the theory.

• S-m-n theoremRelated: The theorem operates on program indices supplied by an effective coding.

• Blum axiomsRelated: Programs need indices so the axioms can state uniform conditions across computations.

• Gödel's first incompleteness theoremRelated: Arithmetic statements can then talk about other arithmetic statements, enabling self-reference.

• Turing's proofRelated: Machine descriptions can be encoded as data so one machine can examine another’s description.