Use of Indiscernibles in Model Theory - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T17:49:43Z http://mathoverflow.net/feeds/question/50305 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/50305/use-of-indiscernibles-in-model-theory Use of Indiscernibles in Model Theory Eran 2010-12-24T21:20:29Z 2010-12-29T16:39:31Z <p>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?</p> <p>Thanks.</p> http://mathoverflow.net/questions/50305/use-of-indiscernibles-in-model-theory/50308#50308 Answer by Andres Caicedo for Use of Indiscernibles in Model Theory Andres Caicedo 2010-12-24T21:40:57Z 2010-12-24T21:40:57Z <p>Eran,</p> <p>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.</p> <p>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 <em>stability theory</em> or (more recently, as in the work of <a href="http://math.yonsei.ac.kr/bkim/" rel="nofollow">Byunghan Kim</a>) of <em>simplicity</em>. The point is that the notions of <em>forking</em> and <em>dividing</em> 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.</p> <p>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 <em>sharps</em>. A good reference for this use is Kanamori's "The higher infinite".</p> http://mathoverflow.net/questions/50305/use-of-indiscernibles-in-model-theory/50667#50667 Answer by Dave Marker for Use of Indiscernibles in Model Theory Dave Marker 2010-12-29T16:39:31Z 2010-12-29T16:39:31Z <p>Some other classical uses of indisceribles due to Morley:</p> <ul> <li><p>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.</p></li> <li><p>If for all $\alpha&lt;\omega_1$ 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 $\alpha&lt;\omega_1$,then it has arbitrarily large models. These results need the Erd\"os-Rado partition theorem.</p></li> </ul>