The stability spectrum for classes of atomic models (Q2909618)

The analysis of models of a sentence \(\phi\) of \(L_{\omega_1, \omega}\) can be profitably translated (for instance, in the study of categoricity in power) to that of models of a first-order theory \(T\) that omit a collection \(\Gamma\) of first-order types over the empty set. In particular, when \(\phi\) is complete, \(\Gamma\) can be taken as the collection of non-principal types and consequently the analysis deals with the atomic models of \(T\).NEWLINENEWLINEIn this general framework the authors consider, for a complete first-order theory \(T\) with monster model \(U\), two notions of Stone spaces of types over an atomic subset \(A\) of \(U\). The former, \(S_{\mathrm{at}}(A)\), is the collection of \(p \in S(A)\) such that, for every tuple \(a \in U\) realizing \(p\), \(Aa\) is also atomic. The latter, \(S_{\mathrm{mod}}(A)\), is the collection of \(p \in S(A)\) such that \(p\) is realized in some atomic model \(M\) of \(T\) extending \(A\).NEWLINENEWLINELet \(i\) denote `at' or `mod', \(K(T)\) be the class of atomic models of \(T\). Say that \(K(T)\) is \(i\)-stable in some infinite cardinal \(\lambda\) if, for every positive integer \(m\) and every \(M \in K(T)\) of power \(\lambda\), \(|S_i^m(M)| = \lambda\). Then \(K(T)\) is called \(i\)-stable if it is \(i\)-stable in some \(\lambda\), \(i\)-superstable if it is \(i\)-stable in every suitably large \(\lambda\), and strictly \(i\)-stable if it is \(i\)-stable but not \(i\)-superstable. Finally, let \(\delta(T)\) denote the well-ordering number of the class of models of \(T\) omitting a family of types, as defined by the second author in his book [Classification theory and the number of non-isomorphic models. Amsterdam etc.: North-Holland Publishing Company (1978; Zbl 0388.03009)]NEWLINENEWLINE NEWLINEThe first main result of the paper regards unstable theories. It says: Assume that for some positive integer \(m\) and for every \(\alpha < \delta(T)\), there is \(M \in K(T)\) for which \(|S_i^m(M)| > |M|^{\beth_\alpha(|T|)}\). Then for every \(\lambda \geq |T|\), there is \(M\) with \(|S_i^m(M)| > |M| = \lambda\).NEWLINENEWLINEThe second main result gives some information on the spectrum of strictly stable theories. It proves the following: Suppose that for every \(\alpha < \delta (T)\), there is \(M \in K(T)\) such that \(|S_i^m(M)| > |M| \geq \beth_\alpha\). Then for any \(\mu\) with \(\mu^{\aleph_0} > \mu\), \(K(T)\) is not \(i\)-stable in \(\mu\).NEWLINENEWLINEThe proof is based on the building of suitable tree indiscernibles. This also leads the authors to discuss the role of various types of index sets for indiscernibles and some variants of the Ehrennfeucht-Mostowski construction.