What is an 'imaginary element' in the sense of model theory?
AAn element that satisfies the axioms of the theory in some models but not others
BAn equivalence class of real tuples under a definable equivalence relation, treated as a named element in the imaginary extension M^eq
CA complex number that encodes relational data in an arithmetic structure
DA non-standard element introduced by ultraproduct constructions that violates the standard axioms
An imaginary element is [ā]_E — the equivalence class of a tuple ā under a definable equivalence relation E. It is 'imaginary' because it lives naturally in the mathematical setting (quotient structure) but is not itself an element of the base model M. The imaginary extension M^eq adds a new sort for each such quotient, making these classes into genuine named objects that can be quantified over and used as parameters.
Question 2 Multiple Choice
In the theory of algebraically closed fields, a definable subgroup H of a definable group G produces cosets that are natural objects of study. Without imaginary elements, a model theorist faces which difficulty?
ANo difficulty — cosets are already elements of the field in any algebraically closed field
BAwkward coding tricks to represent cosets using field elements, since cosets are sets of elements rather than individual elements of the model
CA restriction to studying only finite groups H, where cosets can be enumerated
DAn inability to write first-order formulas defining H or G within the field language
A coset gH is a set of field elements — a subset of the domain, not a single element. Without M^eq, to work with cosets one must encode them indirectly using field elements, often choosing representatives and tracking equivalence by hand. This is cumbersome and breaks the clean model-theoretic machinery (types, definability, independence) that requires objects to be elements. M^eq resolves this by making each coset a genuine element in a new sort.
Question 3 True / False
A theory 'eliminates imaginaries' if every imaginary element is interdefinable with a real tuple — meaning the structure M^eq adds no genuinely new objects beyond those already nameable in M.
TTrue
FFalse
Answer: True
Elimination of imaginaries says: for every definable equivalence relation E on M^n, there is a definable function f: M^n → M^k such that E(ā, b̄) iff f(ā) = f(b̄). In other words, every equivalence class is coded by a real tuple, so no new sort is needed. Algebraically closed fields and strongly minimal theories eliminate imaginaries, which is part of why they have such clean geometric structure — all the definable quotients are already visible within the original model.
Question 4 True / False
Adding imaginary elements to a model typically makes the theory more complex and harder to work with, which is why most model theorists avoid M^eq when possible.
TTrue
FFalse
Answer: False
The opposite is true: M^eq typically makes theories easier to analyze by promoting natural quotient objects to first-class elements. Once cosets, orbits, and other definable quotients are named elements, the full model-theoretic toolkit — types, definability, forking independence — applies directly without coding work-arounds. Theories that eliminate imaginaries (where M and M^eq are essentially equivalent) are considered the cleanest and best-behaved, not because M^eq is avoided, but because it adds nothing genuinely new.
Question 5 Short Answer
Why do model theorists construct M^eq, the imaginary extension of M? What problem does it solve that cannot be resolved within M itself?
Think about your answer, then reveal below.
Model answer: M^eq solves the problem that definable equivalence relations produce natural quotient objects — equivalence classes — that are sets of elements rather than elements of M. A structure can define the relation E(x, y) (saying x and y are equivalent) without being able to name the equivalence classes themselves. This gap means the model-theoretic machinery (quantifying over types, using objects as parameters, studying definability and independence) cannot be applied to these quotient objects directly. M^eq adds a new sort for each definable quotient, making these classes into genuine elements. Now the full toolkit applies, and structural properties like stability, forking, and geometric rank can be analyzed over all definable objects, not just real elements.
The need for M^eq reflects a general phenomenon in logic: a theory can define relations on its domain without being able to name all the sets those relations carve out. Imaginaries fill the gap between 'definable relation' and 'nameable object.'