The ordinal numbers are themselves well-ordered. This would lead to the Burali-Forte paradox, except we work around this by saying the ordinals are a "class" and not a "set."
I'm wondering if an alternative approach exists with anti-foundation axioms.
Within these set theories, is it possible for the set of all ordinals to actually exist, as a true set?
The reason I ask is that with the ordinary Burali-Forti paradox, the set of all ordinals cannot exist, or else it would have to contain itself and hence define a new ordinal. But if ill-founded sets are allowed, there is no problem with the set of ordinals containing itself.
I'm sort of envisioning a structure where, as you define the ordinals, everything is nicely well ordered, well founded, etc. Then, only at the very top, when you want to look at the set of all ordinals, does this set turn out to contain itself and be ill-founded, and hence not define a new ordinal. But I'm not sure if there's just another paradox that emerges then.