Boolean algebras of elementary characteristic (1, 0, 1) whose set of atoms and Ershov–Tarski ideal are computable

Abstract

It is proved that there exists a computable Boolean algebra of elementary characteristics(1, 0, 1) which has a computable set of atoms and a computable Ershov-Tarski ideal, but no strongly computable isomorphic copy. Also a description of Δ 0 6 -computable Boolean algebras is presented. PRELIMINARIES AND BACKGROUNDA model is said to be computable if its universe is a computable set, operations are computable functions, and relations are computable. A computable Boolean algebra is n-computable if there exists an algorithm which, given a finitary Σ n -formula and a tuple of elements, decides whether the formula is true on the tuple. A strongly computable model is one for which such an algorithm exists for all formulas of predicate calculus. A Boolean algebra is decidable if it has a strongly computable isomorphic copy. As a source of preliminary information on the theory of Boolean algebras, we use [1].Following [2], we will conceive of Boolean algebras as models for a language Σ BA = {0, 1, +, ·, −}, where the sum (denoted +), the product (denoted ·), and the difference (denoted −) correspond to union, intersection, and complement, respectively.

Strong Constructivizability of Boolean Algebras of Elementary Characteristic (∞, 0, 0)

“…For this case, in particular, the ultimate answer was given in [9]. In [10][11][12][13][14], answers were found for all possible subsets S ⊆ Ψ n+1 , thereby accomplishing such a long-term pursuit.…”