 Provability logic therefore needs a different approach. Based on a suggestion by J. McKinsey and A. Tarski (1944), L. Esakia (1981, 2003) investigated the interpretation of \(\Diamond\) as the derived set operator \(d\), which maps a set \(B\) to the set of its limit points \(d(B)\). To explain the consequences of this interpretation of \(\Diamond\), we need two more definitions, namely of the concepts dense-in-itself and scattered. A subset \(B\) of a topological space is called dense-in-itself if \(B \subseteq d(B)\). A topological space is called scattered if it has no non-empty subset that is dense-in-itself. The ordinals in their interval topology form examples of scattered spaces. Esakia (1981) proved an important correspondence: he showed that a topological space satisfies the axiom GL if and only if the space is scattered. This correspondence soon led to the result, independently found by Abashidze (1985) and Blass (1990), that provability logic is complete with respect to any ordinal \(\ge \omega^\omega\).