Characterizing all models in infinite cardinalities (Q1935865)

Let \(A(L,\kappa)\) denote the following condition: For any models \(\mathfrak{A}\) and \(\mathfrak{B}\) of cardinality \(\kappa\), if \(\mathfrak{A}\) and \(\mathfrak{B}\) satisfy the same \(L\)-theory then they are isomorphic. Let \(L^n\) denote \(n\)th order logic. It was shown by \textit{M. Ajtai} [Ann. Math. Logic 16, 181--203 (1979; Zbl 0415.03044)] that \(A(L^2,\omega)\) is independent of ZFC. Here the author continues and extends the work of Ajtai. He asks: what kind of a logic \(L\) is needed to characterize all models of cardinality \(\kappa\) (in a finite vocabulary) up to isomorphism by their \(L\)-theories? Of course, \(L_{\kappa^+,\kappa^+}\) is enough. So he looks for small definable logics, i.e., logics for which the sentences are hereditarily of smaller cardinality than \(\kappa\). He shows that for any cardinal \(\kappa\) it is independent of ZFC whether such a small definable logic exists. It can be second-order logic in case \(\kappa = \omega\). For uncountable \(\kappa\) it can be fourth-order logic or a certain infinitary second-order logic \(L^2_{\kappa,\omega}\). He also investigates the role of generalized quantifiers.