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Hi, what is the main use of Indiscernibles in model theory? reading through Chang and Keisler's Model Theory it seems that the main motivation for Indicernibles is for getting many non isomorphic models for a theory (like the theory of dense linear order without endpoint). Also, can you recommend the best source for reading about Indiscernibles and their uses?

Thanks.

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I'm not sure if this is the sort of thing you're interested in, but: a more modern usage of indiscernible sequences is in the study of "nice" (simple, or stable, or . . .) theories via Morley sequences, which are indiscernible sequences satisfying additional properties. For example, in a simple theory, Morley sequences witness interdependencies in a very nice way (see Grossberg, Iovino, Lessman's paper "A Primer of Simple Theories"). –  Noah S Dec 24 '10 at 21:28
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Eran,

As far as I know, indiscernibility is used in two ways in model theory. One, as you say, is to obtain many non-isomorphic models. This is for sure the classical use of indiscernibility.

Another, more modern one, is to have access to tools such as Ramsey's theorem and its uncountable version, the Erdős-Rado theorem. This is useful in some formulations of stability theory or (more recently, as in the work of Byunghan Kim) of simplicity. The point is that the notions of forking and dividing are cleaner to formulate in the presence of sufficiently indiscernible sequences. (So one typically works in large saturated structures in this context.) There are several modern references for stability, etc, where the use of indiscernibility is apparent, see for example Frank Wagner's "Simple theories", Mathematics and its applications, Kluwer Academic Publishers, 2000.

A third use of indiscernibility is fairly common in set theory, where it is the most common approach to defining the large cardinal notions known as sharps. A good reference for this use is Kanamori's "The higher infinite".

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Some other classical uses of indisceribles due to Morley:

  • In the proof that $\kappa$-categorical theories are $\omega$-stable (for $\kappa\ge\aleph_1$), he constructs a model of size $\kappa$ realizing only countably many types over each countable set by taking a model generated by well ordered indiscernibles.

  • If for all $\alpha<\omega_1$ there is a model of size $\beth_\alpha$ omitting a type $p$, then there are arbitrarily large models omitting $p$, or, more generally, if an $L_{\omega_1,\omega}$ sentence has models of size $\beth_\alpha$ for all $\alpha<\omega_1$,then it has arbitrarily large models. These results need the Erd\"os-Rado partition theorem.

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