A collection of sentences in a fixed logical language, typically closed under logical consequence, whose members serve as axioms or consequences that characterize a class of models.

Principle

A theory organizes syntactic commitments: it specifies which sentences are accepted as true and determines the class of structures that satisfy those sentences.

Demonstration

The first-order theory of groups comprises the group axioms; the complete theory of the real closed fields consists of all first-order sentences true in the real numbers under the field language.

Misapplication

Equating a theory with a single finite presentation without acknowledging closure under consequence, or treating a set of informal assumptions as a formal theory, is a misuse.

Consequence

Working with a theory lets one deduce sentences by logical consequence, compare expressiveness, and speak of completeness, decidability, and model classes determined by the theory.

Reversal

Instead of starting from sentences and asking for models, reverse the perspective by starting from a class of structures and asking for the theory of all sentences true in them (the theory of the class).

Boundary

Typically refers to a theory in a specified formal language and logic (e.g., first-order); it excludes informal axiom systems not closed under logical consequence and frameworks requiring higher-order or infinitary languages unless stated.

Semantic Tension

Tension exists between 'axiomatization' (a generating set of sentences) and the whole theory closed under consequence; one emphasizes generators, the other the full closure.

Synthesis

A theory is the syntactic locus of a mathematical subject: a set of sentences closed under consequence that both encodes assumptions and determines the semantic class of models those sentences describe.