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 {{harvtxt|Shelah|1990}}, and elimination of imaginaries was introduced by {{harvtxt|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 M{{space|hair}}n 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
- {{Citation | last1=Hodges | first1=Wilfrid | author1-link=Wilfrid Hodges | title=Model theory | publisher=Cambridge University Press | isbn=978-0-521-30442-9 | year=1993 | url-access=registration | url=https://archive.org/details/modeltheory0000hodg }}
- {{citation|mr=0727805
|last=Poizat|first= Bruno
|title=Une théorie de Galois imaginaire. [An imaginary Galois theory]
|journal=Journal of Symbolic Logic |volume=48 |year=1983|issue= 4|pages= 1151–1170|doi=10.2307/2273680|jstor=2273680 }}
- {{Citation | last1=Shelah | first1=Saharon | author1-link=Saharon Shelah | title=Classification theory and the number of nonisomorphic models | origyear=1978 | publisher=Elsevier | edition=2nd | series=Studies in Logic and the Foundations of Mathematics | isbn=978-0-444-70260-9 | year=1990 | url-access=registration | url=https://archive.org/details/classificationth0092shel }}