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STRONG COMPLETENESS OF PROVABILITY LOGIC FOR ORDINAL SPACES

Published online by Cambridge University Press:  19 June 2017

JUAN P. AGUILERA  and
DAVID FERNÁNDEZ-DUQUE
JUAN P. AGUILERA
Affiliation:
INSTITUT FÜR ALGEBRA UND DISKRETE MATHEMATIK TECHNISCHE UNIVERSITÄT WIEN VIENNA, AUSTRIA E-mail: aguilera@logic.at
DAVID FERNÁNDEZ-DUQUE
Affiliation:
CENTRE INTERNATIONAL DE MATHÉMATIQUES ET D’INFORMATIQUE UNIVERSITY OF TOULOUSE, TOULOUSE, FRANCE and DEPARTMENT OF MATHEMATICS INSTITUTO TECNOLÓGICO AUTÓNOMO DE MÉXICO MEXICO CITY, MEXICO E-mail: david.fernandez@irit.fr
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Abstract

Given a scattered space $\mathfrak{X} = \left( {X,\tau } \right)$ and an ordinal λ, we define a topology $\tau _{ + \lambda } $ in such a way that τ +0 = τ and, when $\mathfrak{X}$ is an ordinal with the initial segment topology, the resulting sequence {τ +λ}λ∈Ord coincides with the family of topologies $\left\{ {\mathcal{I}_\lambda} \right\}_{\lambda\in Ord} $ used by Icard, Joosten, and the second author to provide semantics for polymodal provability logics.

We prove that given any scattered space $\mathfrak{X}$ of large-enough rank and any ordinal λ > 0, GL is strongly complete for τ +λ. The special case where $\mathfrak{X} = \omega ^\omega + 1$ and λ = 1 yields a strengthening of a theorem of Abashidze and Blass.

Keywords

Primary 03F45Secondary 03E1003F1554G12provability logictopological semanticsscattered space

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Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 2 , June 2017 , pp. 608 - 628
Copyright
Copyright © The Association for Symbolic Logic 2017 

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References

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