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I am not convinced the definition is what we want. Wouldn't every cardinal above the least PFA cardinal also be a PFA cardin

Yes, that seems to be right. I suppose we want that the forcing is nontrivial unbounded in kappa. Or we could say in the iteration that if PFA holds at that stage, then we add a Cohen real. (by the way, type four tildes to sign your talk posts.) JDH (talk) 19:00, 27 March 2014 (PDT)