 IFL is more expressive than FO. The following are examples of properties expressible in \(\mathbf{ESO}\) – and therefore in \(\mathbf{IFL}\) – but not in \(\mathbf{FO}\): Dedekind-infinity of the domain, non-completeness of a linear order, ill-foundedness of a binary relation, disconnectedness of a graph, equicardinality of the extensions of two first-order formulas \(\phi(x)\) and \(\psi(x)\), infinity of the extension of a first-order formula \(\phi(x)\), and the topological notion of open set (see, e.g., Hintikka 1996, Väänänen 2007). 