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

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

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

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

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

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

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

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

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.