Transplendent models: expansions omitting a type (Q1762364)

Motivated by proofs of existence of maximal automorphisms of countable arithmetically saturated models, the authors introduce general conditions under which an \(L\)-structure \(M\) can be expanded to a model \(M^+\) of a theory \(T\) in a language \(L'\) extending \(L\), omitting a given \(L'\)-type \(p\) with a finite number of parameters. The condition, named \textit{transplendence}, generalizes the notion of resplendence and involves a rather technical consistency requirement on \(T\) and \(p\). The consistency notion used gives rise to a definition of \textit{closed} Scott set. It is shown that if \(M\) is a countable recursively saturated model of a rich theory and the standard system of \(M\) is closed, then \(M\) is transplendent. It follows that every countable model of a rich theory has an elementary extension which is transplendent. More can be proved for models of PA. Engström and Kaye show that a countable recursively saturated model \(M\) of PA is transplendent iff the standard system of \(M\) is closed; and that the standard system of a transplendent model of PA is a \(\beta_\omega\)-model of second-order arithmetic, i.e. \((\omega, \text{SSy}(M))\prec (\omega, P(\omega))\). This last result implies that there are countable arithmetically saturated models of PA which are not transplendent. The other two new notions introduced in the paper are \textit{subresplendence}, which turns out to be equivalent to recursive saturation, and the corresponding notion of \textit{subtransplendence}. A Scott set \(\mathfrak X\) is a \(\beta\)-model if \((\omega, {\mathfrak X})\prec_{\Sigma_1}(\omega, P(\omega))\). A model \(M\) is \textit{\(\beta\)-saturated} if it is \(\mathfrak X\)-saturated for some \(\beta\)-model \(\mathfrak X\). It is shown that every \(\beta\)-model is subtransplendent, and that a model \(M\) of PA is subtransplendent iff it is \(\beta\)-saturated. Another model-theoretic characterization of subtransplendence for models of PA is also given. A model of PA is \textit{\(\omega\)-correct} if, whenever \(M\prec M^*\) and \(M^*\) is \(\omega\)-saturated, then \((M,\omega)\prec (M^*,\omega)\). It is shown that any transplendent model is \(\omega\)-correct. It is an open question whether all countable recursively saturated \(\omega\)-correct models are transplendent. The paper concludes with an application to nonstandard satisfaction classes. The result is that any transplendent model \(M\) of PA has a full satisfaction class \(S\) such that \(\omega\) is definable in \((M,S)\).