Is there a model theoretic realization of the concept of Arithmetical Hierachy? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T14:41:41Z http://mathoverflow.net/feeds/question/37396 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/37396/is-there-a-model-theoretic-realization-of-the-concept-of-arithmetical-hierachy Is there a model theoretic realization of the concept of Arithmetical Hierachy? Tran Chieu Minh 2010-09-01T16:17:53Z 2012-01-08T23:16:19Z <p>The question I want to ask is close to but not exactly what stated in the title:</p> <p>Fix a language $L$, it is known that a statement $\sigma$ is universal in the language if whenever $M$ satisfies $\sigma$ and $N$ is a substructure of $M$ then $N$ also satisfy $\sigma$. It is also known that a statement $\sigma$ is existential if whenever $M$ satisfies $\sigma$ and $N$ is an extension of $M$ then $N$ also satisfy $\sigma$. </p> <p>I can not find generalization of these criteria for formulas with more quantifiers. I wonder why this is the case?</p> http://mathoverflow.net/questions/37396/is-there-a-model-theoretic-realization-of-the-concept-of-arithmetical-hierachy/37406#37406 Answer by Carl Mummert for Is there a model theoretic realization of the concept of Arithmetical Hierachy? Carl Mummert 2010-09-01T17:48:11Z 2010-09-01T17:48:11Z <p>There is one more well-known equivalence for $\forall \exists$ sentences. </p> <p><strong>Theorem</strong> (Chang-Los-Suszko). A theory $T$ is preserved under taking unions of increasing chains of structures if and only if $T$ is equivalent to a set of $\forall \exists$ sentences.</p> <p>For a proof, see Keisler, "Fundamentals of model theory", <em>Handbook of Mathematical Logic</em>, p. 63.</p> <p>I found a related paper, which is older and doesn't quite answer your question but may be of interest. R.&nbsp;C.&nbsp;Lyndon, "Properties preserved under algebraic constructions", Bull. Amer. Math. Soc. 65 n. 5 (1959), 287-299, <a href="http://projecteuclid.org/euclid.bams/1183523309" rel="nofollow">Project Euclid</a></p> <p>According to that paper, and MathSciNet, a general solution to your question should be contained in H.&nbsp;J.&nbsp;Keisler, "Theory of models with generalized atomic formulas", J. Symbolic Logic v. 25 (1960) 1-26, <a href="http://www.ams.org/mathscinet-getitem?mr=130169" rel="nofollow">MathSciNet</a>, <a href="http://www.jstor.org/stable/2964333" rel="nofollow">JStor</a></p> http://mathoverflow.net/questions/37396/is-there-a-model-theoretic-realization-of-the-concept-of-arithmetical-hierachy/85221#85221 Answer by Goldstern for Is there a model theoretic realization of the concept of Arithmetical Hierachy? Goldstern 2012-01-08T23:16:19Z 2012-01-08T23:16:19Z <p>You mean Keisler's sandwich theorem? Chang-Keisler, "Model Theory", theorem 5.2.8: A theory has a $\Pi_{2n}$-set of axioms if the class of its models is closed under $n$-sandwiches. </p>