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 set of an indiscernible sequence in a model of a given theory, provided certain conditions hold. In many theories, long (even infinite) indiscernible sequences exist in sufficiently rich models; saturated models guarantee the existence of arbitrarily long indiscernibles. Indiscernibles are classified as fully, weakly, or order indiscernibles depending on how the dependence on order and parameters is organized.

Applications: Indiscernibles are a central tool in stability theory and Shelah’s classification theory. They allow the transfer of local information about types to global structural properties, aid in constructing models with prescribed features, and help analyze the complexity of theories by reducing questions to the behavior of indiscernible sequences.

In set theory: The notion extends to indiscernibles for inner models such as L. The existence of a rich set of L-indiscernibles is closely tied to the hypothesis 0#, and such indiscernibles are used to study canonical inner models and large-cardinal phenomena.