 Are all structures second-order characterizable? There are only countably many second-order sentences, hence only countably many (up to isomorphism) second-order characterizable structures. Therefore there are lots of structures of every infinite cardinality which are not second-order characterizable. However, it is not easy to give examples. One example is \((\kappa,&lt;)\), where \(\kappa\) is the first measurable cardinal (\(&gt;\omega\)). See  §7.2  for an explanation. Another example is \((\oN,&lt;,A)\), where A is the set of Gödel numbers of valid second-order sentences in the vocabulary of one binary relation. For an explanation, see  §7.2.  One may ask whether there is an example which arises from mathematical practice outside of mathematical logic. What about \((\oR,+,\times,&lt;,A)\), where A is a Hamel basis (a basis for the vector space where the reals are the vectors, rationals are the scalars and vector addition is the usual addition of reals)? It is consistent, relative to the consistency of ZF that no such structure \((\oR,+,\times,&lt;,A)\) is second-order characterizable (Hyttinen, Kangas, &amp; Väänänen 2013).