# Topological Interpretations of Provability Logic

## Abstract

Provability logic concerns the study of modality \(\Box \) as provability in formal systems such as Peano Arithmetic. A natural, albeit quite surprising, topological interpretation of provability logic has been found in the 1970s by Harold Simmons and Leo Esakia. They have observed that the dual \(\Diamond \) modality, corresponding to consistency in the context of formal arithmetic, has all the basic properties of the topological derivative operator acting on a scattered space. The topic has become a long-term project for the Georgian school of logic led by Esakia, with occasional contributions from elsewhere. More recently, a new impetus came from the study of polymodal provability logic \(\mathbf {GLP}\) that was known to be Kripke incomplete and, in general, to have a more complicated behavior than its unimodal counterpart. Topological semantics provided a better alternative to Kripke models in the sense that \(\mathbf {GLP}\) was shown to be topologically complete. At the same time, new fascinating connections with set theory and large cardinals have emerged. We give a survey of the results on topological semantics of provability logic starting from first contributions by Esakia. However, a special emphasis is put on the recent work on topological models of polymodal provability logic. We also include a few results that have not been published so far, most notably the results of Sect. 10.4 (due to the second author) and Sects. 10.7, 10.8 (due to the first author).

## Keywords

Provability logic Scattered spaces GLP Ordinal topologies## Notes

### Acknowledgments

We wish to thank the referee for many useful comments, which helped to significantly improve the readability of the paper. Thanks are also due to Guram Bezhanishvili both for his detailed comments and his patience with the slow pace this chapter was taking. The first author was supported by the Russian Foundation for Basic Research (RFBR), Russian Presidential Council for Support of Leading Scientific Schools, and the Swiss–Russian cooperation project STCP–CH–RU “Computational proof theory”. The second author was supported by the Shota Rustaveli National Science Foundation grant #FR/489/5-105/11 and the French–Georgian grant CNRS–SRNSF #4135/05-01.

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