Nondiversity in substructures

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 cofinal 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.