Cardinality of total orders on a countably infinite set ($2^{{\aleph }_0}$)

There are at most continuum many binary relations on a countable set. For the matching lower bound, order the blocks $\{2n,2n+1\}$ by their index and independently choose the internal comparison in each block. A binary sequence determines those choices and distinct sequences give distinct labelled total orders. The resulting order types can all coincide; the count concerns relations, not isomorphism classes.