 Second-order logic as a foundation for mathematics focuses on mathematical structures rather than mathematical “objects”, as set theory does. In principle every structure has its own “foundation” based on the special features of the structure. The role of second-order logic is to provide a common framework for this. The intuitive informal picture behind second-order logic on a structure is that there is a domain A of individuals and then separate domains for all n-ary relations for all \(n\in\oN\) and for all n-ary functions for all \(n\in\oN\), associated with A. In comparison to the informal picture of set theory, we have only the domains of subsets, relations and functions, not iterations such as sets of sets, sets of sets of sets and so on, as in set theory. Therefore the intuitive picture behind second-order logic is simpler than the picture behind set theory. In particular, there is no transfinite iteration over all ordinals. Most structures \(\ma\) used in mathematics have a second-order characterization \(\theta_\ma\), see  §10.  Proving (second order) properties \(\phi\) of such structures means deriving \(\phi\) from \(\theta_\ma\). Opinions differ as to whether this derivation should be a syntactic (formal) derivation using some (incomplete) axiom system of second-order logic or whether it should be a semantic derivation (of the form “Every model of the assumptions is a model of the conclusion”). For a working mathematician there is not much difference and probably a semantic derivation is usually favored.