More undecidable lattices of Steinitz exchange systems

We show that the first order theory of the lattice <ω(S) of finite dimensional closed subsets of any nontrivial infinite dimensional Steinitz Exhange System S has logical complexity at least that of first order number theory and that the first order theory of the lattice (S∞) of computably enumerable closed subsets of any nontrivial infinite dimensional computable Steinitz Exchange System S∞ has logical complexity exactly that of first order number theory. Thus, for example, the lattice of finite dimensional subspaces of a standard copy of ⊕ωQ interprets first order arithmetic and is therefore as complicated as possible. In particular, our results show that the first order theories of the lattice (V∞) of c.e. subspaces of a fully effective ℵ0-dimensional vector space V∞ and the lattice of c.e. algebraically closed subfields of a fully effective algebraically closed field F∞ of countably infinite transcendence degree each have logical complexity that of first order number theory.