Independence Structures in Set Theory
Michiel van Lambalgen
Abstract:
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.