In this paper, we discuss a proof system $\mathsf{NGL}$ for the logic $\mathbf{GL}$ of provability, which is equipped with an $ω$-rule. We show the three classes of transitive Kripke frames, the class which strongly validates the $ω$-rule, the class which weakly validates the $ω$-rule, and the class which is defined by the Löb formula, are mutually different, while all of them characterize $\mathbf{GL}$. This gives an example of a proof system $P$ and a class $C$ of Kripke frames such that $P$ is sound with respect to $C$ but the soundness cannot be proved by simple induction on the height of the derivations in $P$. We also show Kripke completeness of $\mathsf{NGL}$ in an algebraic manner. As a corollary, we show that the class of modal algebras which is defined by equations $\Box x\leq\Box\Box x$ and $\bigwedge_{n\inω}\Diamond^{n}1=0$ is not a variety.
In this paper, we investigate Rosser provability predicates whose provability logics are normal modal logics. First, we prove that there exists a Rosser provability predicate whose provability logic is exactly the normal modal logic ${\sf KD}$. Secondly, we introduce a new normal modal logic ${\sf KDR}$ which is a proper extension of ${\sf KD}$, and prove that there exists a Rosser provability predicate whose provability logic includes ${\sf KDR}$.
We investigate several consequences of inclusion relations between quantified provability logics. Moreover, we give a necessary and sufficient condition for the inclusion relation between quantified provability logics with respect to $Σ_1$ arithmetical interpretations.
In this paper, we study the provability logic of intuitionistic theories of arithmetic that prove their own completeness. We prove a completeness theorem for theories equipped with two provability predicates $\Box$ and $\triangle$ that prove the schemes $A\to\triangle A$ and $\Box\triangle S\to\Box S$ for $S\inΣ_1$. Using this theorem, we determine the logic of fast provability for a number of intuitionistic theories. Furthermore, we reprove a theorem previously obtained by M. Ardeshir and S. Mojtaba Mojtahedi determining the $Σ_1$-provability logic of Heyting Arithmetic.
The branch of provability logic investigates the provability-based behavior of the mathematical theories. In a more precise way, it studies the relation between a mathematical theory $T$ and a modal logic $L$ via the provability interpretation which interprets the modality as the provability predicate of $T$. In this paper we will extend this relation to investigate the provability-based behavior of a hierarchy of theories. More precisely, using the modal language with infinitely many modalities, $\{\Box_n\}_{n=0}^{\infty}$, we will define the hierarchical counterparts of some of the classical…
In 1933, Gödel introduced a provability interpretation of the propositional intuitionistic logic to establish a formalization for the BHK interpretation. He used the modal system, $\mathbf{S4}$, as a formalization of the intuitive concept of provability and then translated $\mathbf{IPC}$ to $\mathbf{S4}$. His work suggested the problem to find a concrete provability interpretation of the modal logic $\mathbf{S4}$. In this paper, we will try to answer this problem. In fact, we will generalize Solovay's provability interpretation of the modal logic $\mathbf{GL}$ to capture other modal logics suc…
Assuming the existence of suitable large cardinals, we show it is consistent that the Provability logic $\mathbf{GL}$ is complete with respect to the filter sequence of normal measures. This result answers a question of Andreas Blass from 1990 and a related question of Beklemishev and Joosten.
For the Heyting Arithmetic HA, HA* is defined as the theory $\{A\mid {\sf HA}\vdash A^{\Box}\}$, where $A^{\Box}$ is called the box translation of $A$. We characterize the $Σ_1$-provability logic of HA* as a modal theory ${\sf iH}_σ^*$.
Let $\mathcal{PL}({\sf T},{\sf T}')$ and $\mathcal{PL}_{Σ_1}({\sf T},{\sf T}')$ respectively indicates the provability logic and $Σ_1$-provability logic of ${\sf T}$ relative in ${\sf T}'$. In this paper we characterize the following relative provability logics: $\mathcal{PL}_{Σ_1}({\sf HA},\mathbb{N})$, $\mathcal{PL}_{Σ_1}({\sf HA},{\sf PA})$, $\mathcal{PL}_{Σ_1}({\sf HA}^*,\mathbb{N})$, $\mathcal{PL}_{Σ_1}({\sf HA}^*,{\sf PA})$, $\mathcal{PL}({\sf PA},{\sf HA})$, $\mathcal{PL}_{Σ_1}({\sf PA},{\sf HA})$, $\mathcal{PL}({\sf PA}^*,{\sf HA})$, $\mathcal{PL}_{Σ_1}({\sf PA}^*,{\sf HA})$, $\mathcal…
We introduce the logics GLP(Λ), a generalization of Japaridze's polymodal provability logic GLP(ω) where Λis any linearly ordered set representing a hierarchy of provability operators of increasing strength. We shall provide a reduction of these logics to GLP(ω) yielding among other things a finitary proof of the normal form theorem for the variable-free fragment of GLP(Λ) and the decidability of GLP(Λ) for recursive orderings Λ. Further, we give a restricted axiomatization of the variable-free fragment of GLP(Λ).
Provability logics are modal or polymodal systems designed for modeling the behavior of Gödel's provability predicate in arithmetical theories and its natural extensions. If Λis any ordinal, the Gödel-Löb calculus GLP(Λ) contains one modality [λ] for each λ<Λ, representing provability predicates of increasing strength. GLP(Λ) has no Kripke models, but Beklemishev and Gabelaia recently proved that GLP(ω) is complete for its class of topological models. In this paper we generalize Beklemishev and Gabelaia's result to GLP(Λ) for arbitrary Λ. We also introduce provability ambiances, which are topo…
We consider well-known provability logic GLP. We prove that the GLP-provability problem for variable-free polymodal formulas is PSPACE-complete. For a number n, let L^n_0 denote the class of all polymodal variable-free formulas without modalities <n>, <n+1>,... . We show that, for every number n, the GLP-provability problem for formulas from L^n_0 is in PTIME.
We give a purely syntactical proof of the fixed point theorem for Sacchetti's modal logics ${\bf K} + \Box(\Box^n p \to p) \to \Box p$ ($n \geq 2$) of provability. From our proof, an effective procedure for constructing fixed points in these logics is obtained. We also show the existence of simple fixed points for particular modal formulas.
We prove the uniform interpolation theorem in modal provability logics GL and Grz by a proof-theoretical method, using analytical and terminating sequent calculi for the logics. The calculus for Gödel-Löb's logic GL is a variant of the standard sequent calculus, in the case of Grzegorczyk's logic Grz, the calculus implements an explicit loop-preventing mechanism inspired by work of Heuerding.
In this paper we introduce a modal theory $H_σ$, which is sound and complete for arithmetical $Σ$_1 substitutions in ${\bf HA}$, in other words, we will show that $H_σ$ is the $Σ$_1-provability logic of ${\bf HA}$. Moreover we will show that $H_σ$ is decidable. As a by-product of these results, we show that ${\bf HA} + \Box\bot$ has de Jongh property.
In this paper we consider transfinite provability logics where for each ordinal in some recursive well-order we have a corresponding modal provability operator. The modality [xi] will be interpreted as "provable in ACA_0 together with at most xi nested applications of the omega rule". We show how to formalize this in in second order number theory. Next we prove both soundness and completeness under this interpretation. We conclude by showing how one can lower the base theory ACA_0 to theories below RCA_0.
The provability logic of a theory T is the set of modal formulas, which under any arithmetical realization are provable in T . We slightly modify this notion by requiring the arithmetical realizations to come from a specified set $Γ$. We make an analogous modification for interpretability logics. This is a paper from 2012. We first studied provability logics with restricted realizations, and show that for various natural candidates of theory T and restriction set $Γ$, where each sentence in $Γ$ has a well understood (meta)-mathematical content in T, the result is the logic of linear frames. Ho…
We prove a topological completeness theorem for the modal logic GLP containing operators $\langleλ\rangle$ for $λ\in$ Ord intended to capture progressively stronger notions of consistency in mathematical theories. We show that, given a scattered space $X$ of large-enough rank, any sentence $ϕ$ consistent with GLP can be satisfied in a polytopological space based on the finitely many Icard topologies over $X$ that correspond to the finitely many modalities appearing in $ϕ$.
In this paper, we introduce a proof system $\mathsf{NQGL}$ for a Kripke complete predicate extension of the logic $\mathbf{GL}$, that is, the logic of provability, which is defned by $\mathbf{K}$ and the Löb formula $\Box(\Box p\supset p)\supset\Box p$. $\mathsf{NQGL}$ is a modal extension of Gentzen's sequent calculus $\mathsf{LK}$. Although the propositional fragment of $\mathsf{NQGL}$ axiomatizes $\mathbf{GL}$, it does not have the Löb formula as its axiom. Instead, it has a non-compact rule, that is, a derivation rule with countably many premises. We show that $\mathsf{NQGL}$ enjoys cut ad…
We deal with the fragment of modal logic consisting of implications of formulas built up from the variables and the constant `true' by conjunction and diamonds only. The weaker language allows one to interpret the diamonds as the uniform reflection schemata in arithmetic, possibly of unrestricted logical complexity. We formulate an arithmetically complete calculus with modalities labeled by natural numbers and ω, where ωcorresponds to the full uniform reflection schema, whereas n<ωcorresponds to its restriction to arithmetical Π_{n+1}-formulas. This calculus is shown to be complete w.r.t. a su…
For any ordinal Λ, we can define a polymodal logic GLP(Λ), with a modality [ξ] for each ξ<Λ. These represent provability predicates of increasing strength. Although GLP(Λ) has no Kripke models, Ignatiev showed that indeed one can construct a Kripke model of the variable-free fragment with natural number modalities. Later, Icard defined a topological model for the same fragment which is very closely related to Ignatiev's. In this paper we show how to extend these constructions for arbitrary Λ. More generally, for each Θ,Λwe build a Kripke model I(Θ,Λ) and a topological model T(Θ,Λ), and show th…
The Gödel translation provides an embedding of the intuitionistic logic $\mathsf{IPC}$ into the modal logic $\mathsf{Grz}$, which then embeds into the modal logic $\mathsf{GL}$ via the splitting translation. Combined with Solovay's theorem that $\mathsf{GL}$ is the modal logic of the provability predicate of Peano Arithmetic $\mathsf{PA}$, both $\mathsf{IPC}$ and $\mathsf{Grz}$ admit arithmetical interpretations. When attempting to 'lift' these results to the monadic extensions $\mathsf{MIPC}$, $\mathsf{MGrz}$, and $\mathsf{MGL}$ of these logics, the same techniques no longer work. Following a…
The notion of slow provability for Peano Arithmetic ($\mathsf{PA}$) was introduced by S.D. Friedman, M. Rathjen, and A. Weiermann. They studied the slow consistency statement $\mathrm{Con}_{\mathsf{s}}$ that asserts that a contradiction is not slow provable in $\mathsf{PA}$. They showed that the logical strength of $\mathsf{PA}+\mathrm{Con}_{\mathsf{s}}$ lies strictly between that of $\mathsf{PA}$ and $\mathsf{PA}$ together with its ordinary consistency: $\mathsf{PA}\subsetneq \mathsf{PA}+\mathrm{Con}_{\mathsf{s}}\subsetneq \mathsf{PA}+\mathrm{Con}$. This paper is a further investigation into …
In 1933, Gödel considered two modal approaches to describing provability. One captured formal provability and resulted in the logic GL and Solovay's Completeness Theorem. The other was based on the modal logic S4 and led to Artemov's Logic of Proofs LP. In this paper, we study introduced by the author logic GLA, which is a fusion of GL and LP in the union of their languages. GLA is supplied with a Kripke-style semantics and the corresponding completeness theorem. Soundness and completeness of GLA with respect to the arithmetical provability semantics is established.
In 1994 Jech gave a model theoretic proof of Gödel's second incompleteness theorem for Zermelo-Fraenkel set theory in the following form: ZF does not prove that ZF has a model. Kotlarski showed that Jech's proof can be adapted to Peano Arithmetic with the role of models being taken by complete consistent extensions. In this note we take another step in the direction of replacing proof-theoretic by model-theoretic arguments. We show, without passing through the arithmetized completeness theorem, that the existence of a model of PA of complexity $Σ^0_2$ is independent of PA, where a model is ide…
The provability logic of a theory $T$ captures the structural behavior of formalized provability in $T$ as provable in $T$ itself. Like provability, one can formalize the notion of relative interpretability giving rise to interpretability logics. Where provability logics are the same for all moderately sound theories of some minimal strength, interpretability logics do show variations. The logic IL(All) is defined as the collection of modal principles that are provable in any moderately sound theory of some minimal strength. In this paper we raise the previously known lower bound of IL(All) by…
Beklemishev introduced an ordinal notation system for the Feferman-Schütte ordinal $Γ_0$ based on the autonomous expansion of provability algebras. In this paper we present the logic $\textbf{BC}$ (for Bracket Calculus). The language of $\textbf{BC}$ extends said ordinal notation system to a strictly positive modal language. Thus, unlike other provability logics, $\textbf{BC}$ is based on a self-contained signature that gives rise to an ordinal notation system instead of modalities indexed by some ordinal given a priori. The presented logic is proven to be equivalent to $\textbf{RC}_{Γ_0}$, th…
Provability logic concerns the study of modality $\Box$ 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 $\Diamond$ 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…
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 satisfying all the axioms of GLP are called GLP-spaces. We develop some constructions to build nontrivial GLP-spaces and show that GLP is complete w.r.t. the class of all GLP-spaces.
We consider a many-sorted variant of Japaridze's polymodal provability logic $\mathsf{GLP}$. In this variant, which is denoted $\mathsf{GLP}^\ast$, propositional variables are assigned sorts $α\leq ω$, where variables of finite sort $n < ω$ are interpreted as $Π_{n+1}$-sentences of the arithmetical hierarchy, while those of sort $ω$ range over arbitrary ones. We prove that $\mathsf{GLP}^\ast$ is arithmetically complete with respect to this interpretation. Moreover, we relate $\mathsf{GLP}^\ast$ to its one-sorted counterpart $\mathsf{GLP}$ and prove that the former inherits some well-known prop…
Abashidze and Blass independently proved that the modal logic $\sf{GL}$ 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 $\mathcal I_λ$ for $λ< ω$, with the purpose of providing semantics for Japaridze's polymodal logic $\sf{GLP}$ $_ω$. Icard's construction was later extended by Joosten and the second author to arbitrary ordinals $λ\geq ω$. We further generalize Icard topologies in this article. Given a scattered space $\mathfrak X = (X, τ)$ and an ordinal $λ$, we de…
Strong Completeness of Provability Logic for Uncountable Languages
School of Mathematics, Institute for Research in Fundamental Sciences (IPM)（基础科学研究所数学学院）；Institut für diskrete Mathematik und Geometrie, TU Wien（维也纳工业大学离散数学与几何研究所）
We consider several formalizations in the language of second-order arithmetic of "The formula $ϕ$ is a theorem of $ω$-logic", including some which have been studied in the literature and a new variant defined via a least fixed point. We analyze the provability of relations between these different formalizations in standard theories of reverse mathematics. With this, we study the strength of various reflection principles arising from these notions of provability, surveying known results and establishing some new equivalences, including a characterization of $Π^1_1$-${\sf CA}_0$ in terms of our …
The inverse method is a saturation based theorem proving technique; it relies on a forward proof-search strategy and can be applied to cut-free calculi enjoying the subformula property. Here we apply this method to derive the unprovability of a goal formula G in Intuitionistic Propositional Logic. To this aim we design a forward calculus FRJ(G) for Intuitionistic unprovability which is prone to constructively ascertain the unprovability of a formula G by providing a concise countermodel for it; in particular we prove that the generated countermodels have minimal height. Moreover, we clarify th…
