 If we want to obtain a Downward Löwenheim-Skolem Theorem which works for any structure, not only for structures of a measurable cardinality, we have to look at even bigger cardinals: The LST-number of second-order logic is defined to be the least (if any exist) \(\kappa\) such that if \(\ma\) is any structure and \(\phi\) is a second-order sentence true in \(\ma\), then there is a substructure \(\mb\) of \(\ma\) of cardinality \(\kappa\) which also satisfies \(\phi.\)