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Construction junctions are diamonds. They show where multiple structures must coexist on the same carrier and satisfy compatibility conditions.

Canonical static concept record

Gödel’s first incompleteness theorem

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Summary

Every consistent, effectively axiomatized formal theory strong enough to represent elementary arithmetic is syntactically incomplete.

Record metadata

Canonical name: Gödel’s first incompleteness theorem
Concept ID: first_incompleteness_theorem
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Logic
Domains: Logic, Computability theory
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

effective formal theory containing sufficient arithmetic

Axioms / constraints

consistency or suitable soundness hypotheses, depending on formulation

Canonically induces

An undecidable sentence for the theory

Concept sources

Wikipedia — Gödel's incompleteness theorems — Gödel's incompleteness theorems · encyclopedic reference · source ID wp-j-first-incompleteness
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.

consistent effective theory + arithmetic coding(construction junction)Gödel’s first incompleteness theorem

Permalink to relation

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

Relation ID: j_first_incompleteness__out
Relation type: Combine compatible structures combine-compatible
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: apply Gödel’s theorem

Authored explanation

Under these joint hypotheses, the theory is syntactically incomplete.

How to interpret this relation type

Feed several structures into a construction junction and impose compatibility between them.

Relation sources

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

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

No direct relations are authored in this direction.

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