Models with second order properties IV. A general method and eliminating diamonds

AbstractWe show how to build various models of first-order theories, which also have properties like: tree with only definable branches, atomic Boolean algebras or ordered fields with only definable automorphisms.For this we use a set-theoretic assertion, which may be interesting by itself on the existence of quite generic subsets of suitable partial orders of power λ+, which follows from ♦λ and even weaker hypotheses (e.g., λ=ℵ0, or λ strongly inaccessible). For a related assertion, which is equivalent to the morass see Shelah and Stanley [16].The various specific constructions serve also as examples of how to use this set-theoretic lemma. We apply the method to construct rigid ordered fields, rigid atomic Boolean algebras, trees with only definable branches; all in successors of regular cardinals under appropriate set- theoretic assumptions. So we are able to answer (under suitable set-theoretic assumptions) the following algebraic question.Saltzman's Question. Is there a rigid real closed field, which is not a subfield of the reals