##  [First-Order Theory](/first-order-theory-0) 

 Definition

A set of sentences expressed in first-order logic over a fixed signature, closed under logical consequence, that specifies properties intended to hold in structures of that signature.

 

 

 

 

 

 





## Principle

Principle

A first-order theory organizes mathematical information syntactically: axioms and their consequences in first-order logic determine the class of models; closure under logical consequence ensures the theory contains every sentence provable from its axioms.

 

 

 

 

 





## Demonstration

Demonstration

The axioms of group theory (associativity, identity, inverses) form a first-order theory in the language with a binary symbol · and a constant symbol e; any structure interpreting · and e that satisfies those axioms is a model of the theory, and any sentence derivable from the group axioms belongs to the theory.

 

 

 

 

## Misapplication

Misapplication

Trying to encode inherently second-order properties such as 'the domain is finite' or 'every subset has a maximum' as a first-order theory and expecting the theory to capture them fully; such properties are not preserved under elementary equivalence and so cannot be finitely or axiomatically pinned down in first-order logic.

 

 

 

 

 





## Consequence

Consequence

Given a first-order theory, model-theoretic methods apply: one can study models, elementary extensions, completeness or incompleteness of the theory, and use compactness and Löwenheim–Skolem phenomena to derive existence and size results for models.

 

 

 

 

## Reversal

Reversal

Viewed dually, instead of a deductively closed set of sentences one may start with a class of structures and consider the set of all first-order sentences true in every member; inversion highlights the semantic viewpoint (theory as common theory of a class) rather than the syntactic one.

 

 

 

 

 





## Boundary

Boundary

Applies only to sentences of first-order logic in a fixed signature; excludes higher-order quantification, categorical second-order axioms, and semantic constraints that cannot be expressed or enforced by first-order sentences alone.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension arises between a theory as a purely syntactic, deductively closed object and the semantic class of its models; two different theories can have the same class of models (up to elementary equivalence), and one theory may admit non-isomorphic models sharing all its sentences.

 

 

 

 

 





## Synthesis

Synthesis

A first-order theory is the deductively closed package of first-order sentences in a given signature that describes a collection of structures by specifying what all models must satisfy; it is the bridge from syntactic axioms to semantic model classes within the limits of first-order expressibility.