Independence Results on the Global Structure of the Turing Degrees
Marcia J. Groszek, Theodore A. Slaman · Transactions of the American Mathematical Society · 1983
From $\text {CON(ZFC)}$ we obtain: 1. $\text {CON}(\text {ZFC} + 2^\omega$ is arbitrarily large $+$ there is a locally finite upper semilattice of size ${\omega _2}$ which cannot be embedded into the Turing degrees as an upper semilattice). 2. $\text {CON}(\text {ZFC} + 2^\omega$ is arbitrarily large $+$ there is a maximal independent set of Turing degrees of size ${\omega _1}$).