title: Ehrenfeucht–Mostowski theorem
text: # 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
  
  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
  
  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.
  
  