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Amer Anil Nerode arbitrary distributive lattices coatomless elements coatoms cofinite coinfinite r.e. set construction Contradiction converged by stage COROLLARY countable Boolean algebra cpe(n define definition of s(x densely ordered effectively simple set elements to H exists a recursive filter finite extensions finite sets Friedberg function f hence hyperarithmetical hypersimple set infimum isomorphic Lachlan 11 Lachlan 9 lim x(s,a,n linear order location of marker marker moves Math maximal supersets natural numbers nonrecursive one-to-one recursive function open questions order maximal set ordered basis otherwise pairwise disjoint partial recursive function priority PROOF properties Q.E.D. LEMMA r.e. degrees r.e. set H recursion theory recursive ordinal recursive set recursive subset recursively enumerable sets Rogers 28 s,aX sequence number Stone space subset of NxN substage suffices to show Suppose Theorem Turing degree Turing reducible uniform for finite weak envelope weakly uniform Wf(n Wt(e