A collection of formulas in a formal language taken as a body of statements, often presented as a set of axioms together with all formulas derivable from them; a theory can be finite or infinite, axiomatizable or defined by its set of models.

Principle

A theory organizes knowledge by fixing primitive assumptions (axioms) and closing them under the deductive apparatus of the chosen logic; entailment relates theories to their consequences and to structures that satisfy them.

Demonstration

Group theory as a formal theory consists of the standard group axioms; from these axioms one can deduce theorems such as uniqueness of the identity and properties of inverses, so the axioms plus their consequences constitute the theory of groups.

Misapplication

Confusing an informal, explanatory description with a formal theory (for example, treating an empirical law as an axiom scheme without formalization) leads to category errors about what can be proved inside the system.

Consequence

A well-specified theory yields a clear set of theorems, allows investigation of consistency, completeness, and decidability, and defines classes of structures (models) that realize the theory's claims.

Reversal

Instead of specifying a theory by axioms, one may specify it by its class of models (semantic definition); this reversal highlights the duality between syntactic and semantic characterizations of the same mathematical content.

Boundary

A theory is relative to a chosen formal language and logic; it does not by itself imply uniqueness of models, nor does it guarantee decidability or semantic completeness without further properties.

Semantic Tension

There is tension between a theory viewed as a set of sentences (syntactic object) and as a description of a mathematical subject given by a class of models (semantic object); the two views coincide only under completeness conditions.

Synthesis

A theory is the formal package—axioms plus their deductive closure—that defines a domain of discourse via constraints on models and supplies the inferential apparatus from which theorems are derived and analyzed.