Ehrenfeucht–Mostowski theorem
In model theory, a field within mathematical logic, the Ehrenfeucht–Mostowski theorem (Ehrenfeucht & Mostowski 1956) gives conditions for the existence of a model with indiscernibles.
Statement
[edit]A linearly ordered set X is called a set of indiscernibles of a model if the truth of a statement about elements of X depends only on their order.
The Ehrenfeucht–Mostowski theorem states that if T is a theory with an infinite model, then there is a model of T containing any given linearly ordered set X as a set of indiscernibles.
The proof uses Ramsey's theorem.
Applications
[edit]The Ehrenfeucht–Mostowski is used to construct models with many automorphisms. It is also used in the theory of zero sharp to construct indiscernibles in the constructible universe.
References
[edit]- Ehrenfeucht, A.; Mostowski, A. (1956), "Models of axiomatic theories admitting automorphisms", Polska Akademia Nauk. Fundamenta Mathematicae, 43: 50–68, ISSN 0016-2736, MR 0084456