 The bigger the characterizable structure is, the more complex is the second-order theory. That there is no largest characterizable structure can be seen as follows: If \(\ma\) is second-order characterizable, then so is the reduct of \(\ma\) to the empty  vocabulary[8],  that is, the cardinality \(|A|\) of \(\ma\) is characterizable. Such cardinal numbers were studied by Garland (1974). For example, if \(\kappa\) is characterizable, then so are \(\kappa^+\) and \(2^\kappa\).