Jokes aside, isn't this period of sedevacante does put into question many claims about the Papacy brought forward by the Roman Catholic Church? Like my earlier comment about the Church being built on Peter and that Peter only has one successor, what now? On whom is the Church built on? And how are the Churches of the Catholic Communion in communion with one another? How are they "in communion with Rome" if there is no Bishop of Rome?
One theory I've considered (I forget when, perhaps around the time JPII died) is that when the pope, i.e. the first-ranking bishop, dies, the second ranking bishop (traditionally the EP, back when he was Catholic) automatically becomes the (interim) first-ranking bishop.
But that opens up another can of worms: could he, during that time, order the new election to be cancelled, making himself the (non-interim) pope?