Limitations on the Fraenkel-Mostowski method of independence proofs

Journal of Symbolic Logic 38 (3):416-422 (1973)
The Fraenkel-Mostowski method has been widely used to prove independence results among weak versions of the axiom of choice. In this paper it is shown that certain statements cannot be proved by this method. More specifically it is shown that in all Fraenkel-Mostowski models the following hold: 1. The axiom of choice for sets of finite sets implies the axiom of choice for sets of well-orderable sets. 2. The Boolean prime ideal theorem implies a weakened form of Sikorski's theorem
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DOI 10.2307/2273037
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Akihiro Kanamori (2006). Levy and Set Theory. Annals of Pure and Applied Logic 140 (1):233-252.
David Pincus (1977). Adding Dependent Choice. Annals of Mathematical Logic 11 (1):105-145.

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