 The ranks 0, 1, 2, 3,… are understood as cardinal numbers. For example, if world \(w_1\) has rank 5 and world \(w_2\) rank 10, then \(w_2\) is disbelieved to a degree twice as much as \(w_1\) is disbelieved. Some ranks may even be empty in the sense that there is no world that has a specific rank \(n\). For example, ranks 3 and 4 may be without any possible worlds, while there are possible worlds at ranks 0, 1, 2, and 5. The cardinal interpretation of the ranks is needed for certain arithmetical operations, such as lifting the ranks by a certain cardinal number, applied to all possible worlds of a certain subset. Because of the possibility of empty ranks, a Grovian system of spheres does not define a unique ranking function.