This is question that probably comes up a lot here, but every time I think I have it straight, I find I don't. So allow me to ask it in the following form. Four elementary facts:
(1)ZFC [ if is an inaccessible cardinal, then ( < , > ZFC) ]
(2) The existence of an inaccessible cardinal is consistent with ZFC
(3)not [ZFC ( ZFC is consistent)].
(4) A theory is consistent iff it has a model.
At first glance, (1) and (4) would seem to imply
(*) ZFC ( ZFC is consistent) which of course is rubbish. One possibility for the problem is that perhaps one cannot assert
(4') ZFC (A theory is consistent iff it has a model),
but I am not sure whether one cannot (since the model concept is formalizable in the language of ZFC), and even if one cannot, I am not sure that that would be the main fissure in the faulty conclusion. I suspect there is something much more basic here that I am missing.