EhrenfeuchtMostowski

Ehrenfeucht–Mostowski constructions are a set of techniques in model theory for building models with prescribed indiscernible sequences. Named after Andrzej Mostowski and Donald H. Ehrenfeucht, the method was introduced in the 1960s and has since become a standard tool in the study of structures and theories.

The central idea is that, for a complete theory T in a countable language and a linear order I, one can construct a model M of T that contains a sequence (a_i) indexed by I which is indiscernible over the empty set. The sequence is arranged so that the order type of I is reflected in the pattern of types tp(a_i1, ..., a_in) for increasing indices i1 < ... < in. In other words, the finite subtuples of the indiscernible sequence realize Types that depend only on the order of their indices, not on the specific positions.

Realizing this pattern typically uses a template or blueprint that assigns, to every finite increasing tuple from I, a complete n-type compatible with T. Using the compactness theorem, one can realize all these types simultaneously in a model M ≽ T, producing an EM model with an I-indexed indiscernible set.

Applications of the Ehrenfeucht–Mostowski construction include producing models with long indiscernibles, exploring stability and classification theory, and obtaining existence results for models in various cardinalities. The method also serves as a tool for analyzing how the combinatorial properties of index orders influence the structure and behavior of models of a theory.