Independence Structures in Set Theory1

The axioms for "independent choices" presented in van Lambalgen [1992] are strengthened here, so that they can be seen as introducing a new type of indiscernibles in set theory. The resulting system allows for the construction of natural inner models. The article is organised as follows. Section 1 introduces the axioms, some preliminary lemmas are proved and the relation with the axiom of choice is investigated. Section 0 gives a philosophical motivation for the axioms; the reader who is not interested in such matters can skip this part. In section 2 we compare the structure introduced by the axioms, here called an independence structure, with two constructions from model theory, indiscernibles and minimal sets. Section 3 contains the construction of inner models, while section 4 presents some concluding philosophical remarks.