Order-isomorphic [formula omitted]-orderings in Cohen extensions

Abstract

In this paper we prove that, in the Cohen extension (adding ℵ 2 -generic reals) of a model M of ZFC+CH containing a simplified ( ω 1 , 1 ) -morass, η 1 -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 η 1 -orderings can be extended to an order-isomorphism between the η 1 -orderings in the Cohen extension of M . We use the simplified ( ω 1 , 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 ℵ 2 generic reals for which the only known arguments require transfinite constructions of order type no greater than ω 1 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, η 1 -orderings are order-isomorphic.