A property of a theory saying that every definable set (with parameters) admits a canonical parameter (an 'imaginary' element) in the home sorts or in a definable expansion, so definable quotients are represented by real elements or named sorts.

Principle

Internalizing quotients: elimination of imaginaries replaces externally described definable equivalence classes or quotients by canonical elements of the structure (possibly after adding natural sorts), enabling uniform naming of definable objects and parametrization of families.

Demonstration

Example: in vector-space structures, subspaces and quotient spaces can be represented by points in Grassmannian-like sorts; for many stable theories one can add sorts for certain definable sets so that every definable set acquires a canonical parameter, making Galois-type and binding-group constructions possible inside the model.

Misapplication

Assuming elimination of imaginaries holds in the original language without checking for necessary definable sorts or parameters; or treating canonical parameters as unique without noting choices up to definable equivalence.

Consequence

When EI holds (perhaps after a controlled expansion), one can form canonical parameters for definable sets, simplify the treatment of types over parameters, develop internal Galois theory, and make definable families uniformly analyzable within the model.

Reversal

Failure of EI means some definable equivalence relations have no canonical representative in the home sorts; one must either work with classes as external objects or expand the language with new sorts to capture them.

Boundary

EI concerns definable quotients and canonical parameters relative to a chosen language and set of sorts; it may require adding definable sorts, and holding in a theory does not automatically transfer to arbitrary expansions or reducts without verification.

Semantic Tension

EI vs naming parameters: EI aims to internalize definable objects as parameters, while an alternative is to treat parameters extrinsically or to name representatives ad hoc; tension also exists between EI and countability/size constraints when canonical parameters live in expanded sorts.

Synthesis

Elimination of imaginaries is the principle that all definable objects and quotients can be represented by canonical elements within the structure (possibly after natural expansions), turning external definable classes into internal, uniformly manageable parameters for model-theoretic constructions.