Models with dimension

W.F. Gross
1976 Bulletin of the Australian Mathematical Society  
It is well known that, if AC is assumed, then any vector space has a basis, any independent subset can be extended to a basis, any two bases of a vector space have the same cardinality and a one-one map between independent subsets can be extended to a monomorphism between the subspaces they generate. We follow [7] in using Marsh's notions of minimal formula and dimension to generalize the notion of vector space, and in defining algebraic closure, independence and basis. The models which we
more » ... der obey the Exchange Lemma and if they also have dimension, then one-one maps between independent subsets are elementary. We first show that, if M is a model with dimension and the algebraic closure of any finite subset is finite, then T(M) is K -categorical, and if M is atomic and has dimension, then either M has a finite basis or the algebraic closure of any finite set is finite. We then investigate whether our models have the properties mentioned in the first paragraph in the situations of not assuming AC and of assuming the axiom of choice for sets of finite sets (ACF). Consistency results are established by constructing permutation models of set theory with atoms (ZFA) and then using the Jech-Sochor transfer theorem. Our results are as follows. It is consistent with ZF that there is a vector space with Dedekind domain and bases of incomparable cardinalities. It is consistent with ZF that there is a vector space with Dedekind domain, a basis and a "maximal" independent subset which is not a basis. Without AC one-one maps between
doi:10.1017/s0004972700024953 fatcat:6f5a6m6kjneatprsmf6ls6lvca