 Theorem 26. Upward Löwenheim-Skolem Theorem. Let \(\Gamma\) be any set of sentences of \(\LKe,\) such that for each natural number \(n\), there is an interpretation \(M_n = \langle d_n,I_n\rangle\), such that \(d_n\) has at least \(n\) elements, and \(M_n\) satisfies every member of \(\Gamma\). In other words, \(\Gamma\) is satisfiable and there is no finite upper bound to the size of the interpretations that satisfy every member of \(\Gamma\). Then for any infinite cardinal \(\kappa\), there is an interpretation \(M=\langle d,I\rangle\), such that the size of \(d\) is at least \(\kappa\) and \(M\) satisfies every member of \(\Gamma\).