More undecidable lattices of Steinitz Exchange Systems
We show that the first order theory of the lattice ℒ (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 ℒ(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 ⊕ Q interprets first order arithmetic and is therefore as complicated as possible. In particular, our results show that the first order theories of the lattice ℒ(V ) of c.e. subspaces of a fully effective N -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. <ω ∞ ∞ ω ∞ 0 ∞ ∞
Journal of Symbolic Logic
Galminas, L. R. and Rosenthal, John W., "More undecidable lattices of Steinitz Exchange Systems" (2002). Faculty Articles Indexed in Scopus. 2185.