Imaginary element

In model theory, a branch of mathematics, 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 of that theory 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.

References

• Hodges, Wilfrid (1993), Model theory, Cambridge University Press, ISBN 978-0-521-30442-9, https://archive.org/details/modeltheory0000hodg 
• Poizat, Bruno (1983), "Une théorie de Galois imaginaire. [An imaginary Galois theory]", Journal of Symbolic Logic 48 (4): 1151–1170, doi:10.2307/2273680 
• 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, https://archive.org/details/classificationth0092shel