##  [First-Order Theory](/first-order-theory-1) 

 Definition

A set of first‑order sentences in a fixed signature (vocabulary) — closed or considered as a generating set for its deductive closure — whose intended models are the structures that satisfy every sentence in the set.

 

 

 

 

 

 





## Principle

Principle

Mathematical structures are described by quantification over elements (not over sets or relations) in a fixed signature; the theory collects the sentences that are taken as axioms or constraints for those structures within first‑order logic.

 

 

 

 

 





## Demonstration

Demonstration

The theory of groups is the set of first‑order sentences expressing associativity, identity, and inverses in a language with one binary symbol for multiplication and a constant for the identity; its models are all groups.

 

 

 

 

## Misapplication

Misapplication

Labeling a collection of informal properties as a first‑order theory when some properties require quantification over subsets or functions (second‑order conditions), such as 'every nonempty definable family has a least element' without formal first‑order axiomatization.

 

 

 

 

 





## Consequence

Consequence

A proper first‑order theory can be studied via model‑theoretic tools: satisfiability, compactness, Löwenheim‑Skolem phenomena, completions, and completeness/decidability questions are well‑posed and often tractable within first‑order logic.

 

 

 

 

## Reversal

Reversal

A higher‑order theory permits quantification over relations, functions, or sets and is not confined to first‑order semantics; such theories can express stronger properties but lose certain first‑order meta‑theorems like compactness.

 

 

 

 

 





## Boundary

Boundary

Restricts attention to first‑order languages and sentences; excludes infinitary logics (L_{ω1,ω} etc.), second‑order axioms, and semantic frameworks that quantify over classes or categories of structures.

 

 

 

 

 





## Semantic Tension

Semantic Tension

The term 'theory' sometimes denotes merely a set of axioms, sometimes the deductive closure; the practical tension is between giving a finite axiomatisation and the full (possibly infinite) first‑order theory determined by intended models.

 

 

 

 

 





## Synthesis

Synthesis

A first‑order theory is the collection of first‑order statements in a given signature that collectively specify the properties of interest for a class of structures, forming the basis for formal deduction and model‑theoretic analysis.