Order-isomorphic Morass-definable ${\eta }_1$-orderings

Abstract

We prove that in the Cohen extension adding ${\aleph }_3$ generic reals to a model of $ZFC+CH$ containing a simplified $({\omega }_1,2)$-morass, gap-2 morass-definable ${\eta }_1$-orderings with cardinality ${\aleph }_3$ are order-isomorphic. Hence it is consistent that the $2^{{\aleph }_0}={\aleph }_3$ and that morass-definable ${\eta }_1$-orderings with cardinality of the continuum are order-isomorphic. We prove that there are ultrapowers of $\mathbb{R}$ over $\omega$ 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 ${\omega }_1$, to a function between objects of cardinality ${\aleph }_3$.