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We prove that in the Cohen extension adding $\aleph_3$ generic reals to amodel of $ZFC+CH$ containing a simplified $(\omega_1,2)$-morass, gap-2morass-definable $\eta_1$-orderings with cardinality $\aleph_3$ areorder-isomorphic. Hence it is consistent that the $2^{\aleph_0}=\aleph_3$ andthat morass-definable $\eta_1$-orderings with cardinality of the continuum areorder-isomorphic. We prove that there are ultrapowers of $\mathbb{R}$ over$\omega$ that are gap-2 morass-definable. The constructions use a simplifiedgap-2 morass, and commutativity with morass-maps and morass-embeddings, toextend a transfinite back-and-forth construction of order type $\omega_1$, to afunction between objects of cardinality $\aleph_3$.

Order-isomorphic Morass-definable $\eta_1$-orderings