Finite variable logic stability and finite models (Q2747724)

Let \(L\) be a first-order language, and \(L^n\) the set of formulas with at most \(n\) distinct variables. An \(L^n\)-theory is complete if it implies or refutes all \(L^n\)-sentences; an embedding \(M\hookrightarrow N\), or a substructure \(M\subseteq N\), is \(L^n\)-elementary if \(L^n\)-formulas with parameters in \(M\) are preserved. The author develops basic model and stability theory for complete \(L^n\)-theories. His proofs mainly consist of adaptations or applications of the corresponding first-order results.NEWLINENEWLINENEWLINEA complete \(L^n\)-theory \(T\) has the \(L^n\)-amalgamation property if any two \(L^n\)-elementary models \((M_1,A)\) and \((M_2,A)\), where \(A\) is a common parameter set, can be \(L^n\)-elementarily embedded over \(A\) in a third model of \(T\); it has the \((L^n,\infty)\)-amalgamation property if this is true whenever \(M_1\) and \(M_2\) are infinite. The author shows that \(T\) has the \(L^n\)-amalgamation property iff for any \(\kappa\) every model has an \(L^n\)-elementary \((L^n,\kappa)\)-saturated extension, iff it has an \(L^n\)-elementary strongly \((L^n,\kappa)\)-universal extension (where these are the standard notions with elementary embeddings replaced by \(L^n\)-elementary embeddings). The same theorem holds when restricted to infinite models. Moreover, if \(L\) contains no function symbols and \(n\) is greater or equal to the arity of every relation symbol, then any two \((L^n,\omega,\infty)\)-saturated \(L^n\)-elementarily equivalent models of \(T\) are elementarily equivalent; their (common) theory is called the canonical completion \(T^c\) of \(T\). In particular, in such a model, a type is determined by its restriction to \(L^n\); if \(T\) has only finitely many \(L^n\)-types of arity \(n\) (i.e.\ \(S_n^n(T)\) is finite), then \(T^c\) is \(\omega\)-categorical.NEWLINENEWLINENEWLINEWe call \(T\) stable in \(L^n\) if no \(L^n\)-formula has the order property; it is \(\omega\)-stable in \(L^n\) if there are only countably many \(L^n\)-types over every countable set. Then if \(S_n^n(T)\) is finite, \(T\) is stable in \(L^n\) iff it is \(\omega\)-stable in \(L^n\); if \(L\) has no functions, this is equivalent to \(\omega\)-stability of \(T^c\). Using the corresponding first-order theorem of \textit{B. L. Zil'ber} [Sib. Math. J. 21, 219-230 (1980); translation from Sib. Mat. Zh. 21, No. 2, 98-112 (1980; Zbl 0486.03017); and Sov. Math., Dokl. 24, 149-151 (1981); translation from Dokl. Akad Nauk SSSR 259, 1039-1041 (1981; Zbl 0485.51004)], or \textit{G. Cherlin}, \textit{L. Harrington}, and \textit{A. H. Lachlan} [Ann. Pure Appl. Logic 28, 103-135 (1985; Zbl 0566.03022)], the author concludes that if \(L\) is finite and has no function symbols, and \(M\) is \(\omega\)-stable \(\omega\)-categorical, then for any \(n<\omega\) and any finite set \(A\subset M\) there is a finite \(L^n\)-elementary substructure of \(M\) containing \(A\).NEWLINENEWLINENEWLINEThe author finishes with an example, essentially due to Simon Thomas, where the \(L^n\)-amalgamation property fails.