Abstract

For a model M of Peano Arithmetic, let Lt(M) be the lattice of its elementary substructures, and let Lt⁺(M) be the equivalenced lattice (Lt(M), ≅M), where ≅M is the equivalence relation of isomorphism on Lt(M). It is known that Lt⁺(M) is always a reasonable equivalenced lattice. Theorem. Let L be a finite distributive lattice and let (L,E) be reasonable. If M₀ is a nonstandard prime model of PA, then M₀ has a confinal extension M such that Lt⁺(M) ≅ (L,E). A general method for proving such theorems is developed which, hopefully, will be able to be applied to some nondistributive lattices