Topological completeness of the provability logic GLP

Abstract Provability logic GLP is well-known to be incomplete w.r.t. Kripke semantics. A natural topological semantics of GLP interprets modalities as derivative operators of a polytopological space. Such spaces are called GLP-spaces whenever they satisfy all the axioms of GLP. We develop some constructions to build nontrivial GLP-spaces and show that GLP is complete w.r.t. the class of all GLP-spaces.

[1]  Lev D. Beklemishev,et al.  Topological completeness of the provability logic GLP , 2011 .

[2]  David Fernández-Duque,et al.  Models of transfinite provability logic , 2012, The Journal of Symbolic Logic.

[3]  Leo Esakia,et al.  Intuitionistic logic and modality via topology , 2004, Ann. Pure Appl. Log..

[4]  Lev D. Beklemishev,et al.  Reflection principles and provability algebras in formal arithmetic , 2005 .

[5]  Lev D. Beklemishev.  Veblen hierarchy in the context of provability algebras , 2004 .

[6]  G. Boolos.  The logic of provability , 1993 .

[7]  Lev D. Beklemishev,et al.  Kripke semantics for provability logic GLP , 2007, Ann. Pure Appl. Log..

[8]  Lev D. Beklemishev,et al.  Provability algebras and proof-theoretic ordinals, I , 2001, Ann. Pure Appl. Log..

[9]  Lev D. Beklemishev.  Ordinal Completeness of Bimodal Provability Logic GLB , 2009, TbiLLC.

[10]  David Fernández-Duque,et al.  Turing Progressions and Their Well-Orders , 2012, CiE.

[11]  Thomas Icard,et al.  Models of the Polymodal Provability Logic , 2008 .

[12]  David Gabelaia,et al.  Topological interpretations of provability logic , 2012, 1210.7317.

[13]  Andreas Blass,et al.  Infinitary combinatorics and modal logic , 1990, Journal of Symbolic Logic.

[14]  THOMAS ICARD,et al.  A Topological Study of the Closed Fragment of GLP , 2011, J. Log. Comput..

[15]  David Fernández-Duque,et al.  Kripke Models of Transfinite Provability Logic , 2012, Advances in Modal Logic.

[16]  Guram Bezhanishvili,et al.  On topological models of GLP , 2009 .

[17]  Lev Beklemishev.  On GLP-spaces , 2012 .

[18]  David Fernández-Duque,et al.  The polytopologies of transfinite provability logic , 2012, Arch. Math. Log..

[19]  Guram Bezhanishvili,et al.  Some Results on Modal Axiomatization and Definability for Topological Spaces , 2005, Stud Logica.

[20]  H. Simmons.  Topological aspects of suitable theories , 1975 .

[21]  Konstantin N. Ignatiev,et al.  On strong provability predicates and the associated modal logics , 1993, Journal of Symbolic Logic.