A non structure theorem for an infinitary theory which has the unsuperstability property
Let \(\kappa\), \(\lambda\) be infinite cardinals, \(\psi \in L_{\kappa^+,\omega}\). We say that the sentence \(\psi\) has the \(\lambda\)-unsuperstability property if there are \(\{\phi_ n(\bar x,\bar y):\) \(n<\omega \}\) quantifier free first order formulas in L, a model M of \(\psi\), and there exist \(\{\) \(\bar a_{\eta}:\) \(\eta \in^{\omega \geq}\lambda \}\subseteq | M|\) satisfying: for all \(\eta \in^{\omega}\lambda\), and for every \(\nu \in^{\omega >}\lambda\), \(\nu <\eta \Leftrightarrow M\vDash \phi_{\ell (\nu)}[\bar a_{\nu},\bar a_{\eta}].\) Theorem. Let \(\psi \in L_{\kappa^+,\omega}\), \(\lambda\) a Ramsey cardinal. If \(\psi\) has the \(\lambda\)-unsuperstability property then for every cardinal \(\chi\), \(\chi >| L| \cdot \aleph_ 0\Rightarrow I(\chi,\psi)=2^{\chi}\). A more general theorem is proved; the proof uses a new partition theorem for trees. An application of the theorem to the theory of modules is the following Corollary. Assume there exists a Ramsey cardinal. Let R be an integral domain. If \(DT_ R(=the\) class of torsion divisible R-modules) has a structure theorem (i.e. there are few cardinal invariants such that every module can be characterized by the invariants) then R must be Noetherian. E.g. If every module from \(DT_ R\) is a direct sum of countable generated modules then R is Noetherian.
scientific article; zbMATH DE number 3924763