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Completeness of the Gödel--Löb Provability Logic for Finite Derived Ordinal Topologies
Abstract
We study the Gödel--Löb provability logic 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 U⃗ with oU⃗(κ)≥κ, we obtain a cofinality and GCH preserving generic extension in which Logd(κ,τn)=GL for every n<ω, where τn 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 GLP2 are interpreted respectively by d2k and d2k+1.
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BibTeX
@misc{golshani2026completeness,
title = {{Completeness of the Gödel--Löb Provability Logic for Finite Derived Ordinal Topologies}},
author = {Mohammad Golshani},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29708},
eprint = {2609.29708},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29708},
}
RIS
TY - UNPB
TI - Completeness of the Gödel--Löb Provability Logic for Finite Derived Ordinal Topologies
AU - Mohammad Golshani
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.29708
UR - https://arxiv.org/abs/2609.29708
AB - We study the Gödel--Löb provability logic 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 U⃗ with o^U⃗(κ)≥κ, we obtain a cofinality and GCH preserving generic extension in which Log_d(κ,τ_n)=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 GLP_2 are interpreted respectively by d_2k and d_2k+1.
ER -