indiscernibles
Indiscernibles are a concept in model theory referring to sequences of elements that cannot be distinguished by formulas from a fixed set of parameters. More precisely, in a structure M with a subset A of its domain, a sequence (a_i) indexed by a linear order I is A-indiscernible if, for every natural number n and every two increasing tuples i1 < ... < in and j1 < ... < jn from I, the n-tuples (a_{i1}, ..., a_{in}) and (a_{j1}, ..., a_{jn}) realize the same type over A. If A is empty, the sequence is simply called indiscernible. A sequence is called an order indiscernible when the requirement depends only on the order of indices, not on the specific indices themselves.
Construction and basics: The Ehrenfeucht–Mostowski construction shows how to realize a prescribed ordering as the index
Applications: Indiscernibles are a central tool in stability theory and Shelah’s classification theory. They allow the
In set theory: The notion extends to indiscernibles for inner models such as L. The existence of