Models with second order properties IV. A general method and eliminating diamonds |
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Authors: | Saharon Shelah |
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Institution: | The Institute of Mathematics and the Institute of Advanced Studies, The Hebrew University, Jerusalem, Israel |
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Abstract: | We 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=?0, 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? |
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