I wrote in part:
[...] ask of a topos with [...] whether the negation of X over N (the internal hom [0, X] taken in the slice category over N) can also be occupied.
I have one and a half things backwards here. First of all, of course negation is [X, 0] rather than [0, X]. (But in exponential notation, it is 0^X; that is my excuse.) Also, my placement of "can" implies that the relevant question is whether there ~exists~ a topos E (with given properties) and there exists an object X in E (with the properties that I described); rather, the question is whether for ~every~ E (with given properties) there exists an object X in E (with the properties that I described). Iff so, then the properties required of E are omega-inconsistent. (Iff E must be a terminal category, then they are simply inconsistent. Thus omega-inconsistency is weaker than inconsistency, and omega-consistency is stronger than mere consistency.) --Toby