Models of transfinite provability logic

Fernández–Duque, David; Joosten, Joost J.

doi:10.2178/jsl.7802110

Cited by 30 publications

(53 citation statements)

References 10 publications

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“…David Fernandez and Joost Joosten undertook a thorough study of the variable-free fragment of that logic mostly in connection with the arising ordinal notation systems (see [24,26] for a sample). In particular, they found a suitable generalization of Icard's polytopological space and showed that it is complete for that fragment [25]. Fernandez [29] also proved topological completeness of the full GLP Λ by generalizing the results in [5].…”

Section: Further Resultsmentioning

confidence: 99%

Topological Interpretations of Provability Logic

Beklemishev

Gabelaia

2014

Leo Esakia on Duality in Modal and Intuitionistic Logics

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Provability logic concerns the study of modality ✷ as provability in formal systems such as Peano arithmetic. Natural, albeit quite surprising, topological interpretation of provability logic has been found in the 1970's by Harold Simmons and Leo Esakia. They have observed that the dual ✸ modality, corresponding to consistency in the context of formal arithmetic, has all the basic properties of the topological derivative operator acting on a scattered space. The topic has become a long-term project for the Georgian school of logic led by Esakia, with occasional contributions from elsewhere.More recently, a new impetus came from the study of polymodal provability logic GLP that was known to be Kripke incomplete and, in general, to have a more complicated behavior than its unimodal counterpart. Topological semantics provided a better alternative to Kripke models in the sense that GLP was shown to be topologically complete. At the same time, new fascinating connections with set theory and large cardinals have emerged.We give a survey of the results on topological semantics of provability logic starting from first contributions by Esakia. However, a special emphasis is put on the recent work on topological models of polymodal provability logic. We also included a few results that have not been published so far, most notably the results of Section 6 (due the second author) and Sections 10, 11 (due to the first author).

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Section: Further Resultsmentioning

confidence: 99%

Topological Interpretations of Provability Logic

Beklemishev

Gabelaia

2014

Leo Esakia on Duality in Modal and Intuitionistic Logics

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“…The end-logarithm, last exponent or least exponent ℓ assigns to an ordinal ξ the unique δ such that ξ can be written in the form γ + ω δ ; we also set ℓ0 = 0. Here we see that Exponentials and logarithms shall provide us with key examples of functions to hyperate and cohyperate, respectively, and indeed are the authors' motivation for the present work, due to their applicability to provability logic [5,7].…”

Section: Exponentials and Logarithmsmentioning

confidence: 86%

“…As an academic curiosity by itself it is worthwhile to pursue a notion of transfinite iteration for ordinal functions outright, but the notions of hyperations and cohyperations were originally developed by the authors when working in the field of transfinite provability logics [5,6,7] in order to extend many existing techniques into the transfinite setting. Provability logics, aside from being complex and fascinating on their own right, have been used by Beklemishev to give an ordinal analysis of Peano Arithmetic and related systems [1].…”

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confidence: 99%

Hyperations, Veblen progressions and transfinite iteration of ordinal functions

Fernández–Duque

Joosten

2013

Annals of Pure and Applied Logic

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In this paper we introduce hyperations and cohyperations, which are forms of transfinite iteration of ordinal functions.Hyperations are iterations of normal functions. Unlike iteration by pointwise convergence, hyperation preserves normality. The hyperation f ξ ξ∈On of a normal function f is a sequence of normal functions so that f 0 = id, f 1 = f and for all α, β we have that f α+β = f α f β . These conditions do not determine f α uniquely; in addition, we require that f α α∈On be minimal in an appropriate sense. We study hyperations systematically and show that they are a natural refinement of Veblen progressions.Next, we define cohyperations, very similar to hyperations except that they are left-additive: given α, β, f α+β = f β f α . Cohyperations iterate initial functions which are functions that map initial segments to initial segments. We systematically study co-hyperations and see how they can be employed to define left inverses to hyperations.Hyperations provide an alternative presentation of Veblen progressions and can be useful where a more fine-grained analysis of such sequences is called for. They are very amenable to algebraic manipulation and hence are convenient to work with. Cohyperations, meanwhile, give a novel way to describe slowly increasing functions as often appear, for example, in proof theory.

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“…We will denote the shifted Λ-Icard space on Θ by Ic Θ Λ . The following useful property is a slight modification of a result from [9]:…”

Section: Icard Ambiancesmentioning

confidence: 99%

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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Models of transfinite provability logic

Cited by 30 publications

References 10 publications

Topological Interpretations of Provability Logic

Topological Interpretations of Provability Logic

Hyperations, Veblen progressions and transfinite iteration of ordinal functions

The polytopologies of transfinite provability logic

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