##  [Elimination of Imaginaries](/elimination-imaginaries-0) 

 Definition

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

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

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

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

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

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

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

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

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.