BOOLEAN-ALGEBRAS, STONE SPACES, AND THE ITERATED TURING JUMP

We show, roughly speaking, that it requires omega iterations of the Tu rning jump to decode nontrivial information from Boolean algebras in a n isomorphism invariant fashion. More precisely, if alpha is a recursi ve ordinal, A is a countable structure with finite signature, and d is a degree, we say that A has alphath-jump degree d if d is the least d egree which is the alphath jump of some degree c such there is an isom orphic copy of A with universe omega in which the functions and relati ons have degree at most c. We show that every degree d greater-than-or -equal-to 0(omega) is the omegath jump degree of a Boolean algebra, bu t that for n < omega no Boolean algebra has nth-jump degree d > 0(n). The former results follows easily from work of L. Feiner. The proof of the latter result uses the forcing methods of J. Knight together with an analysis of various equivalences between Boolean algebras based on a study of their Stone spaces. A byproduct of the proof is a method f or constructing Stone spaces with various prescribed properties.