Year: 2026
Type: preprint
Abstract: We study the Gödel--Löb provability logic $\mathbf{GL}$ for finite derived topologies on the ordinals. Assuming GCH and the existence of a measurable cardinal $κ$ which carries a coherent sequence of normal measures $\vec{U}$ with $o^{\vec{U}}(κ)\geqκ$, we obtain a cofinality and GCH preserving generic extension in which $\text{Log}_d(κ,τ_n)=\mathbf{GL}$ for every $n<ω$, where $τ_n$ is the $n$-th derived topology. The result answers a well-know open question. Extending Beklemishev's theorem, we show that in the same model, for every $k<ω$, $ \text{Log}_d(κ;τ_{2k},τ_{2k+1})=\mathbf{GLP}_2, $ where the two modalities of $\mathbf{GLP}_2$ are interpreted respectively by $d_{2k}$ and $d_{2k+1}$.
Source: arXiv (Cornell University)
Author: Golshani, Mohammad
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214366755