 An axiom of infinity can say in second-order logic that a proper subset of the domain has the same cardinality as the entire domain (i.e., that the domain is not Dedekind-finite), or that there is a partial order without a maximal element, or that there is a set with a unary function and a constant which constitute a structure isomorphic to \((\oN,s,0)\), or that the domain is the union of two disjoint sets which have the same cardinality as the domain, and so on. As is the case in set theory without the Axiom of Choice, the different formulations of infiniteness need not be equivalent. In second-order logic the situation is even more diffuse because of the variety of different formulations of the Axiom of Choice. We refer to Asser (1981) for a discussion of the different variants and to Hasenjaeger (1961) for a proof that the various non-equivalent forms of Axioms of Infinity form in a sense a dense set. For a survey of different concepts of finiteness, see de la Cruz (2002).