Henkin constructions of models with size continuum

In the novel White Light [Ruc80], Rudy Rucker proposes a metaphor for the continuum hypothesis. One can reach א1 by a laborious climb up the side of Mt. ON, pausing at 0. Or one can take Cantor’s instantaneous elevator through the center of the mountain. In this paper, working in ZFC, we take Shelah’s elevator, which is a bit slower. After countably many floors, each with finitely many rooms, we reach an object of cardinality 2א0 . The underlying construction applies for finding atomic models, two-cardinal theorems, a collection of continuum many points that are asymptotically similar (a weak form of indiscernibility), and a coloring with a Borel square of size continuum. In his seminal Denumerable models of complete theories, [Vau61], Vaught