Topological Interpretations of Provability Logic

Beklemishev, Lev D.; Gabelaia, David

doi:10.1007/978-94-017-8860-1_10

Cited by 25 publications

(21 citation statements)

References 46 publications

(76 reference statements)

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“…If the inequality is strict, then ξ is a limit point of d β τc Θ, as is it easy to check that {ζ < ξ : cf(ζ) = Ω β } is stationary in ξ (i.e., it intersects every club). By [3,Theorem 11.5], τ c is generated by I 1 and sets {d I1 A : A ⊂ Θ}, so that if, on the contrary,…”

mentioning

confidence: 99%

Strong Completeness of Provability Logic for Ordinal Spaces

Aguilera¹,

Fernández–Duque²

2017

J. symb. log.

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Abashidze and Blass independently proved that the modal logic GL 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 I λ for λ < ω, with the purpose of providing semantics for Japaridze's polymodal logic GLPω. 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 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 I λ when X is an ordinal endowed with the left topology.We then prove that, given any scattered space X and any ordinal λ > 0 such that the rank of (X, τ ) is large enough, GL is strongly complete for τ +λ . One obtains the original Abashidze-Blass theorem as a consequence of the special case where X = ω ω and λ = 1.

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mentioning

confidence: 99%

Strong Completeness of Provability Logic for Ordinal Spaces

Aguilera¹,

Fernández–Duque²

2017

J. symb. log.

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“…With this we may also obtain another useful characterization of Icard topologies from [6]: Lemma 6.4. Given ordinals Θ, ξ,…”

Section: Icard Ambiancesmentioning

confidence: 96%

The polytopologies of transfinite provability logic

Fernández–Duque

2014

Arch. Math. Logic

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Provability logics are modal or polymodal systems designed for modeling the behavior of Gödel's provability predicate and its natural extensions. If Λ is any ordinal, the Gödel-Löb calculus GLPΛ contains one modality [λ] for each λ < Λ, representing provability predicates of increasing strength. GLPω has no Kripke models, but it is sound and complete for its topological semantics, as was shown by Icard for the variable-free fragment and more recently by Beklemishev and Gabelaia for the full logic.In this paper we generalize Beklemishev and Gabelaia's result to GLPΛ for countable Λ. We also introduce provability ambiances, which are topological models where valuations of formulas are restricted. With this we show completeness of GLPΛ for the class of provability ambiances based on Icard polytopologies. * Group for Computational Logic, Universidad de Sevilla, dfduque@us.es Topological semanticsRecall that a topological space is a pair X = X, T where T ⊆ P(X) is a family of sets called 'open' such that 1. ∅, X ∈ T 2. if U, V ∈ T , then U ∩ V ∈ T and Definition 3.1 (d-algebra). A d-algebra over a polytopological space X = X, T λ λ<Λ is a collection of sets A ⊆ P(X) which form a Boolean algebra under the standard set-theoretic operations and such that, whenever λ < Λ and S ∈ A it follows that d λ S ∈ A.Below we introduce ambiances, which will be the basis of our semantics; they are a slight generalizarion of polytopological models, which correspond to the special case where A = P(X).Definition 3.2 (Ambiance). An ambiance is a structure X = X, T , A consisting of a polytopological space equipped with a d-algebra A.

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“…We briefly recall a connection between scattered topological spaces and Esakia frames (cf. [3]). Note that we allow Esakia frames and topological spaces to be empty.…”

Section: Remark Let Us Mention An Example Of L-founded Magari Algebrmentioning

confidence: 99%