##  [Theory](/theory-0) 

 Definition

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

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

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

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

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

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

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

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

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.