Imaginary element

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)

Definitions
  • 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).

Examples
  • 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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See also

References
  • Hodges, Wilfrid (1993), Model theory, Cambridge University Press, ISBN 978-0-521-30442-9
  • Poizat, Bruno (1983), "Une théorie de Galois imaginaire. [An imaginary Galois theory]", Journal of Symbolic Logic, 48 (4): 1151–1170, doi:10.2307/2273680, JSTOR 2273680, MR 0727805
  • Shelah, Saharon (1990) [1978], Classification theory and the number of nonisomorphic models, Studies in Logic and the Foundations of Mathematics (2nd ed.), Elsevier, ISBN 978-0-444-70260-9
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