By Solovay's celebrated completeness result on formal provability we know that the provability logic $\mathrm GL$ describes exactly all provable structural properties for any sound and strong enough arithmetical theory with a decidable axiomatisation. Japaridze generalised this result by considering a polymodal version $\mathrm GLP$ of $\mathrm GL$ with modalities $[n]$ for each natural number $n$ referring to ever increasing notions of provability.
Modern treatments of $\mathrm GLP$ tend to interpret the $[n]$ provability notion as "provable in a base theory $T$ together with all true $Π^0_n$ formulas as oracles". In this paper we generalise this interpretation into the transfinite. In order to do so, a main difficulty to overcome is to generalise the syntactical characterisations of the oracle formulas of complexity $Π^0_n$ to the hyper-arithmetical hierarchy. The paper exploits the fact that provability is $Σ^0_1$ complete and that similar results hold for stronger provability notions. As such, the oracle sentences to define provability at level $α$ will recursively be taken to be consistency statements at lower levels: provability through provability whence the name of the paper.
The paper proves soundness and completeness for the proposed interpretation for a wide class of theories; namely for any theory that can formalise the recursion described above and that has some further very natural properties. Some remarks are provided on how the recursion can be formalised into second order arithmetic and on lowering the proof-theoretical strength of these systems of second order arithmetic.
A formula $ϕ$ is called \emph{$n$-provable} in a formal arithmetical theory $S$ if $ϕ$ is provable in $S$ together with all true arithmetical $Π_{n}$-sentences taken as additional axioms. While in general the set of all $n$-provable formulas, for a fixed $n>0$, is not recursively enumerable, the set of formulas $ϕ$ whose $n$-provability is provable in a given r.e.\ metatheory $T$ is r.e. This set is deductively closed and will be, in general, an extension of $S$. We prove that these theories can be naturally axiomatized in terms of progressions of iterated local reflection principles. In parti…
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.
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.
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 $ϕ$.
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…
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.
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）
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.
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.
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…
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 this paper we present a new proof of Solovay's theorem on arithmetical completeness of Gödel-Löb provability logic GL. Originally, completeness of GL with respect to interpretation of $\Box$ as provability in PA was proved by R. Solovay in 1976. The key part of Solovay's proof was his construction of an arithmetical evaluation for a given modal formula that made the formula unprovable PA if it were unprovable in GL. The arithmetical sentences for the evaluations were constructed using certain arithmetical fixed points. The method developed by Solovay have been used for establishing similar …
