I think the set theory that breaks the Principia Mathematica can be solved by S = ∅ (set of solution is empty).
In case of protest, one should remember that one object lower down the hypothetical chain of sets (by categories) triggers necessary objects all the way up to the "first natural level where one would otherwise see an empty set right below it". "The first natural level" can also be seen as "the deepest level" before, if any at all, the empty set can occur.
You can add all the (meaningless) categories/set containers you want under a natural set/one set that contains members, but where do you get when the bottom container is empty? Clearly, it's just rubbish and thus it's not a serious argument against the project that Principia Mathematica represents.
If this is the only problem, the way for Principia Mathematica is completed. If not, I think there are good chances for completing it.
Your views? Cheers!
[Edit:] Added ""The first natural level" can also be seen as "the deepest level" before, if any at all, the empty set can occur." [End of edit.]
[Edit, 05.03.2011:] Added "You can add all the (meaningless) categories/set containers you want under a natural set/one set that contains members, but where do you get when the bottom container is empty? Clearly, it's just rubbish and thus it's not a serious argument against the project that Principia Mathematica represents." [End of edit.]
