The Omitting Types Theorem is a foundational result of model theory stating that, for a countable first-order language, if a complete type over a theory is not isolated by any single formula, then there is a countable model of the theory that omits that type entirely, meaning no element of the model realizes it. The theorem gives a precise condition for when a description of possible elements can be avoided altogether in some model, complementing the compactness theorem's guarantee that types can generally be realized, and it underlies later model-theoretic constructions such as atomic and prime models.
Facts
StatementIf a complete type p is not isolated then there is a countable model omitting p, provided the language is countable. 1 Classification
Statement Form Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Source Type (model theory) (Wikipedia)
Sources
1. Omitting types theorem, Wikipedia
Omitting types theorem sectionQuote, Omitting types theorem section
The omitting types theorem says that conversely if p is not isolated then there is a countable model omitting p (provided that the language is countable).
Type (model theory) (Wikipedia)
In Branch: Model Theory, Lead sentenceQuote, In Branch: Model Theory, Lead sentence
In model theory and related areas of mathematics, a type is an object that describes how a (real or possible) element or finite co
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