Recall that Turing degrees are equivalence classes of subsets of $\mathbb{N}$ under Turing equivalence (mutual Turing reducibility). They are partially ordered by Turing reducibility and form a join-semilattice of cardinality $2^{\aleph_0}$, which is also known to contain antichains of cardinality $2^{\aleph_0}$. Countable chains can be easily constructed by iterated Turing jump.
Question: Does it contain chains of cardinality $>\aleph_0$?