Support structures for the axiom of choice

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Abstract
The notion of “support” was introduced by Mostowski in [4] in order to prove that a certain universe satisfied the ordering principle but not the axiom of choice. The notion was refined in [3] and in [1] 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 [4] and [1] are characterized. §3 examines a more complicated universe which will be used in [5] to show that in many cases a consistency in full ZF set theory may be proven directly by the methods of [4]. The embedding theorems of §4 are crucial to this application.

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