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I'm working on building a model of ZF where a very weak choice principle fails, in so much as in any permutation model with a set of atoms it is inconsistent that this principle fails.

To get a feel for the sort of results around this sort of construction can someone point me to permutation models in the literature which use a class-sized group?

I'm aware of Blass' paper on SVC, but little else.

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Jech's The Axiom of Choice has a couple class permutation models, a review shows theorem 11.2, as well Problems 9.3, 9.4 which you may want to examine (they talk about Injection Principle and Surjection principles from classes to sets, and the failure of them)

Also, while not permutation per se, Monro constructs via symmetric class forcing a Dedekind-finite proper class (that is a class that every function into $\omega$ is bounded). You may find this proof in Independence results concerning Dedekind-finite sets (J. Austral. Math. Soc. 19 (1975), 35–46).

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  • $\begingroup$ I also meant to ask about symmetric models with class-sized group, but some other stuff I read made me think that actually I was asking the wrong questions. $\endgroup$
    – David Roberts ♦
    Commented Dec 19, 2011 at 3:18

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