Order-isomorphic η1-orderings in Cohen extensions

In this paper we prove that, in the Cohen extension (adding ℵ2-generic reals) of a model M of ZFC+CH containing a simplified (ω1,1)-morass, η1-orderings without endpoints having cardinality of the continuum, and satisfying specified technical conditions, are order-isomorphic. Furthermore, any order-isomorphism in M between countable subsets of the η1-orderings can be extended to an order-isomorphism between the η1-orderings in the Cohen extension of M. We use the simplified (ω1,1)-morass, and commutativity conditions with morass maps on terms in the forcing language, to extend countable partial functions on terms in the forcing language that are forced in all generic extensions to be order-preserving injections. This technique provides for the construction of functions in Cohen extensions adding ℵ2 generic reals for which the only known arguments require transfinite constructions of order type no greater than ω1 in models of ZFC+CH. The specific example presented in this paper is an extension of Tarski’s classic result that in models of ZFC+CH, η1-orderings are order-isomorphic.