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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

Model of a theory

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Summary

A first-order structure satisfying every sentence of a specified theory.

Record metadata

Canonical name: Model of a theory
Concept ID: model_of_theory
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.

Carrier(s)

first-order theory $T$ in signature $L$
$L$ -structure $M$

Axioms / constraints

$M$ satisfies every sentence of $T$

Notes

The theory and structure are independent inputs that must use the same signature.

Concept sources

Wikipedia — Model theory — Model theory · expository · source ID wp-model-theory
nLab — model theory — model theory · mathematical reference · source ID nlab-model-of-theory
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.

consistent first-order theory(construction junction)Model of a theory

Permalink to relation

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

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

Authored explanation

Gödel’s completeness theorem guarantees a model for the consistent first-order theory represented by the junction.

How to interpret this relation type

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

Relation sources

Wikipedia — Gödel's completeness theorem — Gödel's completeness theorem · encyclopedic reference · source ID wp-godel-completeness-theorem
nLab — completeness theorem — completeness theorem · mathematical reference · source ID nlab-completeness-theorem
Open Logic Project — Models and Theories — Open Logic Project: Models and Theories · open textbook · source ID openlogic-model-theory

theory + structure satisfying it(construction junction)Model of a theory

Permalink to relation

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

Relation ID: j_model_of_theory__out
Relation type: Combine compatible structures combine-compatible
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: impose satisfaction

Authored explanation

Require $M\models\varphi$ for every $\varphi\in T$ .

How to interpret this relation type

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

Relation sources

Wikipedia — Model theory — Model theory · expository · source ID wp-model-theory
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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Model of a theory

Summary

Record metadata

Carrier(s)

Axioms / constraints

Notes

Concept sources

Incoming relations (arrows to this concept)

consistent first-order theory(construction junction)Model of a theory

Authored explanation

How to interpret this relation type

Relation sources

theory + structure satisfying it(construction junction)Model of a theory

Authored explanation

How to interpret this relation type

Relation sources

Outgoing relations (arrows from this concept)