Abstract

We show that in the lattice $\mathscr{E}_{\Pi}$ of $\Pi^0_1$ classes there are initial segments [$\emptyset$, P] = $\mathscr{L}$ which are not Boolean algebras, but which have a decidable theory. In fact, we will construct for any finite distributive lattice L which satisfies the dual of the usual reduction property a $\Pi^0_1$ class P such that L is isomorphic to the lattice $\mathscr{L}$*, which is $\mathscr{L}$, modulo finite differences. For the 2-element lattice, we obtain a minimal class, first constructed by Cenzer, Downey, Jockusch and Shore in 1993. For the simplest new $\Pi^0_1$ class P constructed, P has a single, non-computable limit point and $\mathscr{L}$* has three elements, corresponding to $\emptyset$, P and a minimal class P$_0 \subset$ P. The element corresponding to $P_0$ has no complement in the lattice. On the other hand, the theory of $\mathscr{L}$ is shown to be decidable. A $\Pi^0_1$ class P is said to be decidable if it is the set of paths through a computable tree with no dead ends. We show that if P is decidable and has only finitely many limit points, then $\mathscr{L}$* is always a Boolean algebra. We show that if P is a decidable $\Pi^0_1$ class and $\mathscr{L}$ is not a Boolean algebra, then the theory of $\mathscr{L}$interprets the theory of arithmetic and is therefore undecidable.