Seems to me like the initial causing of the problem would have to be a one-time self-destructive event. Like maybe we blow up the moon or something and then don't survive the aftermath. We can't use tech to solve the problem if the problem is that we're extinct.

I think that's pretty far out from the sort of thing that people are thinking of when they talk about things you can and can't do with technology, so I think it's more sensible to just say that the category is empty.