 The idea that mathematical concepts are defined implicitly by a set of axioms was proposed by Hilbert but is not uncontroversial (see entry on the  Frege-Hilbert controversy).  The fact that the definition is implicit entails that we only have examples of what sets are without the possibility to formulate any positive predicate that defines them. Elements of a set are not necessarily physical, nor abstract, nor spatial or temporal, nor simple, nor real. The only prerequisite is the possibility to formulate clear judgments about membership. This implicit definition of the notion of a set is not unproblematic. We might define objects that at first glance seem to be proper sets, which after scrutiny appear to be internally inconsistent. This is the basis for: 