Characterizing the Computable Structures: Boolean Algebras and Linear Orders

A countable structure (with finite signature) is computable if its universe can be identified with ω in such a way as to make the relations and operations computable functions. In this thesis, I study which Boolean algebras and linear orders are computable. Making use of Ketonen invariants, I study the Boolean algebras of low Ketonen depth, both classically and effectively. Classically, I give an explicit characterization of the depth zero Boolean algebras; provide continuum many examples of depth one, rank ω Boolean algebras with range ω + 1; and provide continuum many examples of depth ω, rank one Boolean algebras. Effectively, I show for sets S ⊆ ω+1 with greatest element, the depth zero Boolean algebras Bu(S) and Bv(S) are computable if and only if S \{ω} is Σn7→2n+3 in the Feiner Σ-hierarchy. Making use of the existing notion of limitwise monotonic functions and the new notion of limit infimum functions, I characterize which shuffle sums of ordinals below ω + 1 have computable copies. Additionally, I show that the notions of limitwise monotonic functions relative to 0′ and limit infimum functions coincide.