Gap-2 morass-definable η1-orderings

We prove that in the Cohen extension adding ℵ₃ generic reals to a model of $\mathrm{ZFC}+\mathrm{CH}$ containing a simplified (ω₁, 2)-morass, gap-2 morass-definable η₁-orderings with cardinality ℵ₃ are order-isomorphic. Hence it is consistent that $2^{{\aleph }_0}={\aleph }_3$ and that morass-definable η₁-orderings with cardinality of the continuum are order-isomorphic. We prove that there are ultrapowers of $R$ over ω that are gap-2 morass-definable. The constructions use a simplified gap-2 morass, and commutativity with morass-maps and morass-embeddings, to extend a transfinite back-and-forth construction of order-type ω₁ to an order-preserving bijection between objects of cardinality ℵ₃.