Independence results on the global structure of the Turing degrees

Abstract

From CON(ZFC) \text {CON(ZFC)} we obtain: 1. CON ( ZFC + 2 ω \text {CON}(\text {ZFC} + 2^\omega is arbitrarily large + + there is a locally finite upper semilattice of size ω 2 {\omega _2} which cannot be embedded into the Turing degrees as an upper semilattice). 2. CON ( ZFC + 2 ω \text {CON}(\text {ZFC} + 2^\omega is arbitrarily large + + there is a maximal independent set of Turing degrees of size ω 1 {\omega _1} ).