Imaginary element

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In mathematical model theory, an imaginary element of a structure is roughly a definable equivalence class. These were introduced by Shelah (1990), and elimination of imaginaries was introduced by Poizat (1983)


  • M is a model of some theory.
  • x and y stand for n-tuples of variables, for some natural number n.
  • An equivalence formula is a formula φ(x,y) that is a symmetric and transitive relation. Its domain is the set of elements a of Mn such that φ(a,a); it is an equivalence relation on its domain.
  • An imaginary element a/φ of M is an equivalence formula φ together with an equivalence class a.
  • M has elimination of imaginaries if for every imaginary element a/φ there is a formula θ(x,y) such that there is a unique tuple b so that the equivalence class of a consists of the tuples x such that θ(x,b)
  • A model has uniform elimination of imaginaries if the formula θ can be chosen independently of a.
  • A theory has elimination of imaginaries if every model does (and similarly for uniform elimination).


  • ZFC set theory has elimination of imaginaries.
  • Peano arithmetic has uniform elimination of imaginaries.
  • A vector space of dimension at least 2 over a finite field with at least 3 elements does not have elimination of imaginaries.

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