Independence Results on the Global Structure of the Turing Degrees

From CON(ZFC) we obtain: 1. CON(ZFC + 2" is arbitrarily large + there is a locally finite upper semilattice of size W2 which cannot be embedded into the Turing degrees as an upper semilattice). 2. CON(ZFC + 2" is arbitrarily large + there is a maximal independent set of Turing degrees of size Xl). Introduction. Let 6D denote the set of Turing degrees ordered under the usual Turing reducibility, viewed as a partial order K 6D, -) or an upper semilattice K 6D, , V) depending on context. A partial order K A, ?) [upper semilattice KA, , V)] is embeddable into 6D (denoted A -* 6D) if there is an embedding f: A -* 6D so that a ? b if and only iff(a) < f(b) [andf(a) V f(b) = f(a V b)]. The structure of 6D was first investigted in the germinal paper of Kleene and Post [2] where A -* 6D for any countable partial order A was shown. Sacks [9] proved that A -* 6D for any partial order A which is locally finite (any point of A has only finitely many predecessors) and of size at most 2w, or locally countable and of size at most (01. Say X C 6D is an independent set of Turing degrees if, whenever x0, x,. .. ,Xn X and x0 < xi V x2 V Vxn, there is an i between l and n so that x0 = xi; X is maximal if no proper extension of X is independent. Sacks showed (also in [9]) that no countable set of Turing degrees is maximal independent and that there is an independent set of Turing degrees of size continuum. Sacks conjectured that in fact any maximal independent set of Turing degrees must have size continuum. In [11], S. Simpson pointed out that MA(K) implies that every maximal independent set has size greater than K, but suggested that the existence of a maximal independent set of size less than continuum might be independent of ZFC. Theorem 2 confirms Simpson's conjecture. Received by the editors March 30, 1982. 1980 Mathematics Subject Classification. Primary 03D30.