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Canonical static concept record

Compactness theorem

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Summary

A set of first-order sentences has a model whenever every finite subset has a model.

Record metadata

Canonical name: Compactness theorem
Concept ID: compactness_theorem
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Logic
Domains: Logic
Alternate terminology: No alternate terminology is currently authored for this record.
Concept type: Metatheorem

Carrier(s)

set $T$ of first-order sentences

Axioms / constraints

every finite subset of $T$ is satisfiable

Canonically induces

satisfiability of $T$

Notes

The statement is specific to first-order compactness; stronger logics can fail compactness.

Concept sources

Wikipedia — Compactness theorem — Compactness theorem · encyclopedic reference · source ID wp-compactness-theorem
nLab — compactness theorem — compactness theorem · mathematical reference · source ID nlab-compactness-theorem
Open Logic Project — Models and Theories — Open Logic Project: Models and Theories · open textbook · source ID openlogic-model-theory

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Gödel completeness theorem Compactness theorem

Permalink to relation

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

Relation ID: completeness_theorem_to_compactness
Relation type: Theorem implication theorem-implication
Direction: source → target
Endpoint roles: source: Implies by theorem; target: Follows by theorem from
Authored annotation: finite proofs yield compactness

Authored explanation

If an unsatisfiable theory entailed a contradiction, completeness would give a finite formal proof using only finitely many premises; contraposition yields compactness.

How to interpret this relation type

Record a genuine theorem implication that is not part of the target definition; these edges may point toward a weaker structure.

Relation sources

Wikipedia — Compactness theorem — Compactness theorem · encyclopedic reference · source ID wp-compactness-theorem
nLab — compactness theorem — compactness theorem · mathematical reference · source ID nlab-compactness-theorem
Open Logic Project — Models and Theories — Open Logic Project: Models and Theories · open textbook · source ID openlogic-model-theory

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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