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The Σ1-provability logic of HA
Abstract
In this paper we introduce a modal theory iHσ which is sound and complete for arithmetical Σ1-interpretations in HA, in other words, we will show that iHσ is the Σ1-provability logic of HA. Moreover we will show that iHσ is decidable. As a by-product of these results, we show that HA+□⊥ has de Jongh property.
... Here, we consider iSL, the minimal intuitionistic modal logic with both the Gödel-Löb axiom and the completeness axiom, which can also be axiomatised over intuitionistic modal logic iK by the Strong Löb axiom ( ϕ → ϕ) → ϕ. The logic iSL is the provability logic of an extension of Heyting Arithmetic with respect to so-called slow provability [46] and plays an important role in the Σ 1 -provability logic of HA [6]. ...
... Still, they correspond to traditional Hilbert calculi when restricted to consecutions of the shape ∅ ⊢ ϕ, as we do here. Thus, we can connect the generalised Hilbert calculus here to the traditional Hilbert calculus considered by Ardeshir and Mojtahedi [6]. ...
... We now present the Kripke semantics for iSL [34,6] to notably prove soundness of our sequent calculus G4iSLt, and explain its rules (SLtR) and ( →L). The Kripke semantics of iSL is a restriction of the Kripke semantics for intuitionistic modal logics. ...
We provide a new sequent calculus that enjoys syntactic cut-elimination and strongly terminating backward proof search for the intuitionistic Strong L\"ob logic , an intuitionistic modal logic with a provability interpretation. A novel measure on sequents is used to prove both the termination of the naive backward proof search strategy, and the admissibility of cut in a syntactic and direct way, leading to a straightforward cut-elimination procedure. All proofs have been formalised in the interactive theorem prover Coq.
Our paper studies what one might call “the reverse mathematics of explicit fixed points”. We discuss two methods of constructing such fixed points for formulas whose principal connective is the intuitionistic Lewis arrow . Our main motivation comes from metatheory of constructive arithmetic, but the systems in question allow several natural kinds of semantics. The first of these methods, inspired by de Jongh and Visser, turns out to yield a modal system , extending the “gathering” axiom with the standard (“box”) version of the Löb axiom. The second one, inspired by de Jongh and Sambin, seemingly simpler, leads to a modal theory , which proves harder to axiomatize in an elegant way. Apart from showing that both theories are mutually incomparable, we axiomatize their join and investigate several subtheories, whose axioms are obtained as fixed points of simple formulas. We also show that both and are extension stable, that is, their validity in the corresponding preservativity logic of a given arithmetical theory transfers to its finite extensions.
The -provability logic of Peano Arithmetic , is characterized by (Visser, 1982) as , the Gödel-Löb logic plus the completeness principle for atomic variables. Also the -provability logic of the Heyting Arithmetic , is characterized by (Ardeshir & Mojtahedi, 2018) as (for definition of , see Sect. 4.7). In this paper, we find some translation , which embeds in , the intuitionistic counterpart of .
We provide a new sequent calculus that enjoys syntactic cut-elimination and strongly terminating backward proof search for the intuitionistic Strong Löb logic , an intuitionistic modal logic with a provability interpretation. A novel measure on sequents is used to prove both the termination of the naive backward proof search strategy, and the admissibility of cut in a syntactic and direct way, leading to a straightforward cut-elimination procedure. All proofs have been formalised in the interactive theorem prover Coq.
We derive an intuitionistic version of G\"odel-L\"ob modal logic () in the style of Simpson, via proof theoretic techniques. We recover a labelled system, , by restricting a non-wellfounded labelled system for to have only one formula on the right. The latter is obtained using techniques from cyclic proof theory, sidestepping the barrier that 's usual frame condition (converse well-foundedness) is not first-order definable. While existing intuitionistic versions of are typically defined over only the box (and not the diamond), our presentation includes both modalities. Our main result is that coincides with a corresponding semantic condition in birelational semantics: the composition of the modal relation and the intuitionistic relation is conversely well-founded. We call the resulting logic . While the soundness direction is proved using standard ideas, the completeness direction is more complex and necessitates a detour through several intermediate characterisations of .
This paper characterizes the admissible rules for six interesting intuitionistic modal logics: iCK4, iCS4≡IPC, strong Löb logic iSL, modalized Heyting calculus mHC, Kuznetsov-Muravitsky logic KM, and propositional lax logic PLL. Admissible rules are rules that can be added to a logic without changing the set of theorems of the logic. We provide a Gentzen-style proof theory for admissibility that combines methods known for intuitionistic propositional logic and classical modal logic. From this proof theory, we extract bases for the admissible rules, i.e., sets of admissible rules that derive all other admissible rules. In addition, we show that admissibility is decidable for these logics.
Let and respectively indicate the provability logic and -provability logic of T relative in . In this paper we characterise the following relative provability logics: , , , , , , , , , , , (see Table 9.3). It turns out that all of these provability logics are decidable. The notion of reduction for provability logics, first informally considered in (Ardeshir and Mojtahedi 2015). In this paper, we formalize a generalization of this notion (Definition 9.4.1) and provide several reductions of provability logics (see Diagram 9.5). The interesting fact is that is the hardest provability logic: the arithmetical completenesses of all provability logics listed above, as well as well-known provability logics like , , , and , are all propositionally reducible to the arithmetical completeness of .
In this paper, we show how to construct for a given consistent theory U a Σ10-predicate that both satisfies the Löb Conditions and the Kreisel Condition—even if U is unsound. We do this in such a way that U itself can verify satisfaction of an internal version of the Kreisel Condition.
The logic iGLC is the intuitionistic version of Löb's Logic plus the completeness principle A→□A. In this paper, we prove an arithmetical completeness theorems for iGLC for theories equipped with two provability predicates □ and △ that prove the schemes A→△A and □△S→□S for S∈Σ1. We provide two salient instances of the theorem. In the first, □ is fast provability and △ is ordinary provability and, in the second, □ is ordinary provability and △ is slow provability.
Using the second instance, we reprove a theorem previously obtained by Mohammad Ardeshir and Mojtaba Mojtahedi [1] determining the Σ1-provability logic of Heyting Arithmetic.
We show that the provability logic of PA, GL and the truth provability logic, i.e. the provability logic of PA relative to the standard model ℕ, GLS are reducible to their Σ1-provability logics, GLV and GLSV, respectively, by only propositional substitutions.
In the first part of the paper we discuss some conceptual problems
related to the notion of proof. In the second part we survey five major
open problems in Provability Logic as well as possible directions for
future research in this area.
We prove that all extensions of Heyting Arithmetic with a logic that has the finite frame property possess the de Jongh property. Dedicated to Petr Hájek, on the occasion of his 70th Birthday 1. Preface The three authors of this paper have enjoyed Petr Hájek’s acquaintance since the late eighties, when a lively community interested in the metamathematics of arithmetic shared ideas and traveled among the beautiful cities of Prague, Moscow, Amsterdam, Utrecht, Siena, Oxford and Manchester. At that time, Petr Hájek and Pavel Pudlák were writing their landmark book Metamathematics of First-Order Arithmetic [HP91], which Petr Hájek tried out on a small group of eager graduate students in Siena in the months of February and March 1989. Since then, Petr Hájek has been a role model to us in many ways. First of all, we have always been impressed by Petr’s meticulous and clear use of correct notation, witness all his different types of dots and corners, for example in the Tarskian ‘snowing’-snowing lemmas [HP91]. But also as a human being, Petr has been a role model by his example of living in truth, even in averse circumstances [Hav89]. The tragic story of the Logic Colloquium 1980, which was planned to be held in Prague and of which Petr Hájek was the driving force, springs to mind [DvDLS82]. Finally, we were moved by Petr’s open-mindedness when coming to terms with a situation that turned out to look disconcertingly unlike the ‘standard model ’ 1. Therefore, in this paper, we would like to pay homage to Petr Hájek. Unfortunately, we cannot hope to emulate his correct use of dots and corners. Instead, we do our best to provide some pleasing non-standard models and non-classical arithmetics. 2.
In this paper we study NNIL, the class of formulas of the Intuitionistic Propositional Calculus. IPC with no nestings of implications to the left. We show that the formulas of this class are precisely the formulas of the language of IPC that are preserved under taking submodels of Kripke models for IPC (for various notions of submodel). This makes NNIL an analogue of the purely universal formulas in Predicate Logic. We prove a number of interpolation properties for NNIL, and explore the extent to which these properties can be generalized to more complicated classes of formulas.
We prove that Basic Arithmetic, BA, has the de Jongh property, i.e., for any propositional formula A(p 1,..., p n ) built up of atoms p 1,..., p n , BPC ⊢ A(p 1,..., p n ) if and only if for all arithmetical sentences B 1,..., B n , BA ⊢ A(B 1,..., B n ). The technique used in our proof can easily be applied to some known extensions of BA.
0. Introduction.- 1. The Incompleteness Theorems.- 2. Self-Reference.- 3. Things to Come.- 4. The Theory PRA.- 5. Encoding Syntax in PRA.- 6. Additional Arithmetic Prerequisites.- I. The Logic of Provability.- 1. Provability as Modality.- 1. A System of Basic Modal Logic.- 2. Provability Logic(s).- 3. Self-Reference in PRL.- 4. Avoiding R2.- 2. Modal Model Theory.- 1. Model Theory for BML.- 2. Model Theory for PRL.- 3. Models and Self-Reference.- 4. Another Provability Logic.- 3. Arithmetic Interpretations of PRL.- 1. Solovay's First Completeness Theorem.- 2. Solovay's Second Completeness Theorem.- 3. Generalisations, Refinements, and Analogues.- II. Multi-Modal Logic and Self-Reference.- 4. Bi-Modal Logics and Their Arithmetic Interpretations.- 1. Bi-Modal Self-Reference.- 2. Kripke Models.- 3. Carlson Models.- 4. Carlson's Arithmetic Completeness Theorem.- 5. Fixed Point Algebras.- 1. Boolean and Diagonalisable Algebras.- 2. Fixed Point Algebras.- 3. Discussion.- III. Non-Extensional Self-Reference.- 6. Rosser Sentences.- 1. Modal Systems for Rosser Sentences.- 2. Arithmetic Interpretations.- 3. Inequivalent Rosser Sentences.- 7. An Ubiquitous Fixed Point Calculation.- 1. An Ubiquitous Fixed Point Calculation.- 2. Applications.- 3. Relativisation to a Partial Truth Definition.- 4. Svejdar's Self-Referential Formulae.
If Σ is any standard formal system adequate for recursive number theory, a formula (having a certain integer q as its Gödel number) can be constructed which expresses the proposition that the formula with Gödel number q is provable in Σ. Is this formula provable or independent in Σ? [2].
One approach to this problem is discussed by Kreisel in [4]. However, he still leaves open the question whether the formula ( Ex ) ( x, a ), with Gödel-number a, is provable or not. Here ( x, y ) is the number-theoretic predicate which expresses the proposition that x is the number of a formal proof of the formula with Gödel-number y .
In this note we present a solution of the previous problem with respect to the system Z μ [3] pp. 289–294, and, more generally, with respect to any system whose set of theorems is closed under the rules of inference of the first order predicate calculus, and satisfies the subsequent five conditions, and in which the function ( k, l ) used below is definable.
The notation and terminology is in the main that of [3] pp. 306–326, viz. if is a formula of Z μ containing no free variables, whose Gödel number is a, then ({ }) stands for ( Ex ) ( x, a ) (read: the formula with Gödel number a is provable in Z μ ); if is a formula of Z μ containing a free variable, y say, ({ }) stands for ( Ex ) ( x, g ( y )}, where g ( y ) is a recursive function such that for an arbitrary numeral the value of g ( ) is the Gödel number of the formula obtained from by substituting for y in throughout. We shall, however, depart trivially from [3] in writing ( ), where is an arbitrary numeral, for ( Ex ) { x , ).
In this paper extensions of HA are studied that prove their own completeness, i.e. they prove A → □ A, where □ is interpreted as provability in the theory itself. Motivation is three-fold: (1) these theories are thought to have some intrinsic interest, (2) they are a tool for producing and studying provability principles, (3) they can be used to proved independence results. Work done in the paper connected with these motivations is respectively: 1.(i) A characterization is given of theories proving their own completeness, including an appropriate conservation result.2.(ii) Some new provability principles are produced. The provability logic of HA is not a sublogic of the of PA. A provability logic plus completeness theorem is given for a certain intuitionistic extension of HA. De Jongh's theorem for propositional logic is a corollary.3.(iii) FP-realizability in Beeson's proof that KLS is replaced by theories proving their own completeness. New consequences are , .
A system has the existence-property for abstracts (existence property for numbers, disjunction-property) if whenever 'formula presented' for someabstract 'formula presented' for some numeral n; if whenever 'formula presented' A, B are closed). We show that the existence-property for numbers andthe disjunction property are never provable in the system itself; more strongly, the (classically) recursive functions that encode these properties are not provably recursive functions of the system. It is however possible for a system (e.g., 'formula presented') to prove the existenceproperty for abstracts for itself.
We consider interpretations of modal logic in Peano arithmetic (P) determined by an assignment of a sentencev
* ofP to each propositional variablev. We put (⊥)*=“0 = 1”, (χ → ψ)* = “χ* → ψ*” and let (□ψ)* be a formalization of “ψ)* is a theorem ofP”. We say that a modal formula, χ, isvalid if ψ* is a theorem ofP in each such interpretation. We provide an axiomitization of the class of valid formulae and prove that this class is recursive.
This paper is concerned with notions of consequence. On the one hand, we study admissible consequence, specifically for substitutions of Σ10-sentences over Heyting arithmetic (HA). On the other hand, we study preservativity relations. The notion of preservativity of sentences over a given theory is a dual of the notion of conservativity of formulas over a given theory. We show that admissible consequence for Σ10-substitutions over HA coincides with NNIL-preservativity over intuitionistic propositional logic (IPC). Here NNIL is the class of propositional formulas with no nestings of implications to the left. The identical embedding of IPC-derivability (considered as a preorder and, thus, as a category) into a consequence relation (considered as a preorder) has in many cases a left adjoint. The main tool of the present paper will be an algorithm to compute this left adjoint in the case of NNIL-preservativity. In the last section, we employ the methods developed in the paper to give a characterization the closed fragment of the provability logic of HA.
PA is Peano arithmetic. The formula is a formalization of the assertion that the theory interprets the theory (the variables and are intended to range over codes of sentences of PA). We extend Solovay's modal analysis of the formalized provability predicate of PA, , to the case of the formalized interpretability relation . The relevant modal logic, in addition to the usual provability operator `', has a binary operator `' to be interpreted as the formalized interpretability relation. We give an axiomatization and a decision procedure for the class of those modal formulas that express valid interpretability principles (for every assignment of the atomic modal formulas to sentences of PA). Our results continue to hold if we replace the base theory PA with Zermelo-Fraenkel set theory, but not with Godel-Bernays set theory. This sensitivity to the base theory shows that the language is quite expressive. Our proof uses in an essential way earlier work done by A. Visser, D. de Jongh, and F. Veltman on this problem.
A miracle happens. In one hand we have a class of marvelously complex theories in predicate logic, theories with 'sufficient coding potential', like PA (Peano Arithmetic) or ZF (Zermelo Fraenkel Set Theory). In the other we have certain modal propositional theories of striking simplicity. We translate the modal operators of the modal theories to certain specic, fixed, defined predicates of the predicate logical theories. These special predicates generally contain an astronomical number of symbols. We interpret the propositional variables by arbitrary predicate logical sentences. And see: the modal theories are sound and complete for this interpretation. They codify precisely the schematic principles in their scope. Miracles do happen ....
This paper contains the following results: (i) a theorem of the form: if HA (Heyting's Arithmetic) proves some Σ01 substitution instance of an intuitionistically non valid propositional formula then HA proves a substitution instance of a simpler intuitionistically non-valid formula - unless of course the original formula was - in some appropriate sense - already as simple as possible. The result is shown to be adequate. a proof that De Jongh's Completeness Theorem for arithmetical interpretations of Intuitionistic Propositional Logic is verifiable in HA + con(HA). (iii) a characterization of the closed fragment of the provability logic of HA - this is a solution of Friedman's 35th problem for the case of HA. These results are instances of or corollaries to answers of a common kind of question, which we call the evaluation problem for a certain set of interpretations. A framework is developped to analyze this kind of question.
Any recursively enumerable extension of intuitionistic arithmetic which obeys the disjunction property obeys the numerical existence property. Any recursively enumerable extension of intuitionistic arithmetic proves its own disjunction property if and only if it proves its own inconsistency.
Constructivism in Mathematics, vol. I
- Troelstra
Applications of Kripke models
- Smoryński