Relative separation theorems for \(\mathcal L _{\kappa^ + \kappa}\)
This paper strengthens earlier separation theorems for \(L_{\kappa^ + \kappa}\) expressed in terms of an infinite quantifer language \(M_{\lambda^ + \lambda}\). This language allows infinite alternation between conjunction, disjunction and universal existential quantifiers. A consequence of this feature is that the notion of satisfaction has to be defined in terms of infinite games with two players \(\forall\) and \(\exists\); the relation \({\mathcal M}\models\varphi\) means that player \(\exists\) has a winning strategy in the related games. Since such games can be nondetermined, one has to formulate the interpolation theorems as separation theorems. The main result is the following. Assume that \(\varphi\) and \(\psi\) are sentences of \(L_{\kappa^ + \kappa}\) which have no models in common. Then for \(\lambda= \kappa^{<\kappa}\) there is a sentence \(\theta\) of \(M_{\lambda^ + \lambda}\) in which only such nonlogical symbols occur which occur both in \(\varphi\) and \(\psi\) and were for all \({\mathcal M}\), \[ \begin{aligned} (i)\;\;\;&\quad \text{if } {\mathcal M} \models \varphi, \text{ then } {\mathcal M} \models \theta,\\ (ii)\;\;&\quad \text{if }\forall \text{ does not have a winning strategy in the game of } {\mathcal M} \text{ and } \theta, \text{ then } {\mathcal M} \not\models \varphi.\\ \text{Notice that \((ii)\) implies} (iii)\;&\quad \text{if } {\mathcal M}\models \theta, \text{ then } {\mathcal M}\not\models \psi\end{aligned} \] which corresponds to the strength of earlier separation results. The author has also analogous versions of the interpolation theorems of Lyndon and Malitz. He also improves earlier results on the undefinability of well-orderings.
scientific article; zbMATH DE number 120373