Strong Completeness of Provability Logic for Ordinal Spaces

2015/11/18 by Aguilera, Juan P., Fernández-Duque, David · 1 citation
#FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.1511.05882

Abstract

Abashidze and Blass independently proved that the modal logic \sfGL is complete for its topological interpretation over any ordinal greater than or equal to ω^ω equipped with the interval topology. Icard later introduced a family of topologies \mathcal I_λ for λ< ω, with the purpose of providing semantics for Japaridze's polymodal logic \sfGLP _ω. Icard's construction was later extended by Joosten and the second author to arbitrary ordinals λ≥ ω. We further generalize Icard topologies in this article. Given a scattered space \mathfrak X = (X, τ) and an ordinal λ, we define a topology τ_+λ in such a way that τ₊₀ is the original topology τ and τ_+λ coincides with \mathcal I_λ when \mathfrak X is an ordinal endowed with the left topology. We then prove that, given any scattered space \mathfrak X and any ordinal λ>0 such that the rank of (X, τ) is large enough, \sfGL is strongly complete for τ_+λ. One obtains the original Abashidze-Blass theorem as a consequence of the special case where \mathfrak X=ω^ω and λ=1.

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• Provability Models

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