The notion of “support” was introduced by Mostowski in  in order to prove that a certain universe satisfied the ordering principle but not the axiom of choice. The notion was refined in  and in  it was shown to be satisfied in a certain Cohen model of full ZF set theory. This paper is an axiomatic study of universes whose undefined relations are ∈ and a “support structure”, T.
In §2 the general theory is introduced and the universes of  and  are characterized. §3 examines a more complicated universe which will be used in  to show that in many cases a consistency in full ZF set theory may be proven directly by the methods of . The embedding theorems of §4 are crucial to this application.