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2. The model with atoms. We use the original permutation model of Fraenkel. This model, and variants of it, is discussed by Mostowski in [5]. We do not give the construction of the model here. Instead we list some statements which are true in the model and base our subsequent arguments on these statements. The statements follow easily from observations of [5] and [6]. 1. The axioms of ZFA hold. 2. There is a function P on co such that the Pt are mutually disjoint pairs and K = U t eœ Pt is the set of atoms. 3. There is a relationV(n, x), also written xe V„, which satisfies: (a) Every set is in some V„, n e co. (b) Each Vw contains all ordinals, the function P, and the members of

Independence of the prime ideal theorem from the Hahn Banach theorem