A property of a theory that every definable equivalence class (an 'imaginary') can be coded by a tuple of real elements (elements of the home sorts), so that every imaginary has a canonical parameter in the real sorts and no extra imaginary sorts are needed.

Principle

The organizing rule is that definable quotients should admit definable representatives in ordinary sorts: for any definable set X and definable equivalence relation E on X there is a definable function f from X into some M^n such that x E y iff f(x)=f(y), giving canonical parameters for E‑classes.

Demonstration

Given a definable equivalence relation E on a definable set X, elimination of imaginaries supplies a definable coding f:X→M^n. For example, in a structure with named tuples one can directly take f to return a chosen tuple representative; more theoretically, many well‑behaved theories can be shown to eliminate imaginaries after adding finitely many canonical parameters or auxiliary sorts.

Misapplication

Confusing elimination of imaginaries with the stronger claim that every definable set is interdefinable with a single real tuple, or neglecting that some theories require adding finitely many sorts to achieve elimination; assuming EI holds automatically can produce incorrect identifications of canonical parameters.

Consequence

EI simplifies the bookkeeping of definability and canonical parameters, enables a cleaner treatment of imaginaries in stability and simplicity theory, and often makes internality and analysability arguments more transparent.

Reversal

Failure of EI means there exist definable equivalence classes that cannot be coded by real tuples; one then either works with added imaginary sorts or studies weaker forms like weak elimination of imaginaries or elimination of finite imaginaries.

Boundary

A notion about first‑order definability and definable quotients; it may require introducing finitely many new sorts to hold, and it is distinct from elimination of hyperimaginaries (which concerns equivalence relations type‑definable over parameters rather than definable).

Semantic Tension

Tension appears between EI and weaker notions (weak EI, elimination of finite imaginaries) and between working in the home sorts versus enriching the language with sorts; deciding which route to take affects canonical parameter arguments and convenience.

Synthesis

Elimination of imaginaries demands that every definable quotient be representable by a real tuple, giving canonical parameters inside the ordinary sorts and streamlining definability and classification arguments by avoiding the proliferation of imaginary sorts.