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Canonical static concept record

Gödel numbering

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Summary

An effective coding of symbols, formulas, and finite proofs by natural numbers so that syntactic operations become arithmetically representable.

Record metadata

Canonical name: Gödel numbering
Concept ID: godel_numbering
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Computability theory
Domains: Computability theory, Logic, Number theory
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

formal syntax
natural numbers

Data

effective encoding

Canonically induces

Arithmetization of syntax

Concept sources

Wikipedia — Gödel numbering — Gödel numbering · encyclopedic reference · source ID wp-godel-numbering
nLab — Gödel numbering — Gödel numbering · mathematical reference · source ID nlab-godel-numbering
Gödel — Über formal unentscheidbare Sätze — Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I · original research paper · source ID godel-incompleteness-paper
Open Logic Project — Incompleteness — Incompleteness · open textbook · source ID openlogic-incompleteness

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Formal proof system Gödel numbering

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: story_formal_proof_system_to_godel_numbering_add_data
Relation type: Add data add-data
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: choose an effective arithmetical coding

Authored explanation

For an effectively presented formal system, a Gödel numbering adds a computable encoding of symbols, formulas, and finite proofs by natural numbers.

How to interpret this relation type

Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.

Relation sources

Gödel — Über formal unentscheidbare Sätze — Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I · original research paper · source ID godel-incompleteness-paper

Natural numbers $\mathbb N$ Gödel numbering

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: story_natural_numbers_to_godel_numbering_add_data
Relation type: Add data add-data
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: choose an effective coding of syntax

Authored explanation

Use natural numbers as codes and choose a computable encoding of symbols, formulas, and finite proofs; Gödel numberings are not unique.

How to interpret this relation type

Relation sources

Gödel — Über formal unentscheidbare Sätze — Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I · original research paper · source ID godel-incompleteness-paper
Open Logic Project — Incompleteness — Incompleteness · open textbook · source ID openlogic-incompleteness

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.