Topological Completeness of the Transfinite Provability Logic
Juan P. Aguilera · arXiv (Cornell University) · 2016
Let $\Lambda$ be an ordinal. The polymodal provability logic GLP$_\Lambda$ contains modalities $\langle\lambda\rangle$ for $\lambda < \Lambda$ intended to capture progressively stronger notions of consistency in mathematical theories. We show GLP$_\Lambda$ is complete with respect to its topological interpretation, where each modality $\langle \xi \rangle$ denotes the derived-set operator in some topology $\mathcal{T}_\xi$. Specifically, for each ordinal $\Lambda$ and any tall-enough scattered space $(X, \tau)$, one defines topologies that are to $\tau$ as 'iterated order topologies' are to the ordinals. We show that, if we restrict the domain of valuations or further refine the topologies, then the logic of the resulting polytopological space is GLP$_\Lambda$.