Boolean algebra approximations

Abstract

Knight and Stob proved that every low 4 _4 Boolean algebra is 0 ( 6 ) 0^{(6)} -isomorphic to a computable one. Furthermore, for n = 1 , 2 , 3 , 4 n=1,2,3,4 , every low n _n Boolean algebra is 0 ( n + 2 ) 0^{(n+2)} -isomorphic to a computable one. We show that this is not true for n = 5 n=5 : there is a low 5 _5 Boolean algebra that is not 0 ( 7 ) 0^{(7)} -isomorphic to any computable Boolean algebra. It is worth remarking that, because of the machinery developed, the proof uses at most a 0 ′ ′ 0^{\prime \prime } -priority argument. The technique used to construct this Boolean algebra is new and might be useful in other applications, such as to solve the low n _n Boolean algebra problem either positively or negatively.