Imaginary element
Encyclopedia
In mathematical model theory
, an imaginary element of a structure is roughly a definable equivalence class.
These were introduced by , and elimination of imaginaries was introduced by
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
, an imaginary element of a structure is roughly a definable equivalence class.
These were introduced by , and elimination of imaginaries was introduced by
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.