I’ll make this more explicit, and note that at present I don’t see it affecting anyone’s argument:
ZF can’t prove its own self-consistency and thus can’t prove it has a model. You logician says that’s as it should be, since not every model of ZF contains another model—which seems true provided we treat each model more as an interpretation than as an object in a larger universe—and then says that a model has to be “well-populated”—which AFAICT holds for standard models, but not for all other interpretations like those your logician used in the previous sentence.
I’ll make this more explicit, and note that at present I don’t see it affecting anyone’s argument:
ZF can’t prove its own self-consistency and thus can’t prove it has a model. You logician says that’s as it should be, since not every model of ZF contains another model—which seems true provided we treat each model more as an interpretation than as an object in a larger universe—and then says that a model has to be “well-populated”—which AFAICT holds for standard models, but not for all other interpretations like those your logician used in the previous sentence.