A proof of the consistency of Heyting arithmetic formulated in natural deduction is given. The proof is a reduction procedure for derivations of falsity and a vector assignment, such that each reduction reduces the vector. By an interpretation of the expressions of the vectors as ordinals each derivation of falsity is assigned an ordinal less than ε 0, thus proving termination of the procedure.

We present an extension of Heyting arithmetic in finite types called Uniform Heyting Arithmetic that allows for the extraction of optimized programs from constructive and classical proofs. The system has two sorts of first-order quantifiers: ordinary quantifiers governed by the usual rules, and uniform quantifiers subject to stronger variable conditions expressing roughly that the quantified object is not computationally used in the proof. We combine a Kripke-style Friedman/Dragalin translation which is inspired by work of Coquand and (...) Hofmann and a variant of the refined A-translation due to Buchholz, Schwichtenberg and the author to extract programs from a rather large class of classical first-order proofs while keeping explicit control over the levels of recursion and the decision procedures for predicates used in the extracted program. (shrink)

We define classes Φnof formulae of first-order arithmetic with the following properties:(i) Everyφϵ Φnis classically equivalent to a Πn-formula (n≠ 1, Φ1:= Σ1).(ii)(iii)IΠnandiΦn(i.e., Heyting arithmetic with induction schema restricted to Φn-formulae) prove the same Π2-formulae.We further generalize a result by Visser and Wehmeier. namely that prenex induction within intuitionistic arithmetic is rather weak: After closing Φnboth under existential and universal quantification (we call these classes Θn) the corresponding theoriesiΘnstill prove the same Π2-formulae. In a second part (...) we consideriΔ0plus collection-principles. We show that both the provably recursive functions and the provably total functions ofare polynomially bounded. Furthermore we show that the contrapositive of the collection-schema gives rise to instances of the law of excluded middle and hence. (shrink)

In this paper, we give a constructive nonstandard model of intuitionistic arithmetic (Heyting arithmetic). We present two axiomatisations of the model: one finitary and one infinitary variant. Using the model these axiomatisations are proven to be conservative over ordinary intuitionistic arithmetic. The definition of the model along with the proofs of its properties may be carried out within a constructive and predicative metatheory (such as Martin-Löf's type theory). This paper gives an illustration of the use of (...) sheaf semantics to obtain effective proof-theoretic results.The axiomatisations of nonstandard intuitionistic arithmetic (to be calledHAIandHAIωrespectively) as well as their model are based on the construction in [5] of a sheaf model for arithmetic using a site of filters. In this paper we present a “minimal” version of this model, built instead on a suitable site of provable filter bases. The construction of this site can be viewed as an extension of the well-known construction of the classifying topos for a geometric theory which uses “syntactic sites”. (Such sites can in fact be used to prove semantical completeness of first order logic in a strictly constructive framework, see [6].)We should mention that for classical nonstandard arithmetics there are several nonconstructive methods of proving conservativity over arithmetic, e.g. the compactness theorem, Mac Dowell–Specker's theorem [3]. (shrink)

In this paper, we give a constructive nonstandard model of intuitionistic arithmetic (Heyting arithmetic). We present two axiomatisations of the model: one finitary and one infinitary variant. Using the model these axiomatisations are proven to be conservative over ordinary intuitionistic arithmetic. The definition of the model along with the proofs of its properties may be carried out within a constructive and predicative metatheory (such as Martin-Löf's type theory). This paper gives an illustration of the use of (...) sheaf semantics to obtain effective proof-theoretic results.The axiomatisations of nonstandard intuitionistic arithmetic (to be calledHAIandHAIωrespectively) as well as their model are based on the construction in [5] of a sheaf model for arithmetic using a site of filters. In this paper we present a “minimal” version of this model, built instead on a suitable site of provable filter bases. The construction of this site can be viewed as an extension of the well-known construction of the classifying topos for a geometric theory which uses “syntactic sites”. (Such sites can in fact be used to prove semantical completeness of first order logic in a strictly constructive framework, see [6].)We should mention that for classical nonstandard arithmetics there are several nonconstructive methods of proving conservativity over arithmetic, e.g. the compactness theorem, Mac Dowell–Specker's theorem [3]. (shrink)

Since in Heyting Arithmetic all atomic formulas are decidable, a Kripke model for HA may be regarded classically as a collection of classical structures for the language of arithmetic, partially ordered by the submodel relation. The obvious question is then: are these classical structures models of Peano Arithmetic ? And dually: if a collection of models of PA, partially ordered by the submodel relation, is regarded as a Kripke model, is it a model of HA? Some (...) partial answers to these questions were obtained in [6], [3], [1] and [2]. Here we present some results in the same direction, announced in [7]. In particular, it is proved that the classical structures at the nodes of a Kripke model of HA must be models of IΔ1 and that the relation between these classical structures must be that of a Δ1-elementary submodel. MSC: 03F30, 03F55. (shrink)

Let HA be Heyting's arithmetic, and let CS denote the conjunction of Kreisel's axioms for the creative subject: \begin{eqnarray*} {\rm CS}_1.&&\quad \,\forall\, x (\qed_x A \vee \; \neg \qed_x A)\; ,\nn {\rm CS}_2. &&\quad \,\forall\, x (\qed_x A\to A)\; ,\nn {\rm CS}_3^{\rm S}. &&\quad A\to\,\exists\, x \qed_x A\; ,\nn {\rm CS}_4.&&\quad \,\forall\, x\,\forall\, y (\qed_x A & y \ge x\to\qed_y A)\; .\nn \end{eqnarray*} It is shown that the theory HA + CS with the induction schema restricted to arithmetical (...) (i.e. not containing $\qed$ ) formulas is conservative over HA. (shrink)

In this note we compare propositional logics for closed substitutions and propositional logics for open substitutions in constructive arithmetical theories. We provide a strong example where these logics diverge in an essential way. We prove that for Markov's Arithmetic, that is, Heyting's Arithmetic plus Markov's principle plus Extended Church's Thesis, the logic of closed and the logic of open substitutions are the same.

The subject of this article is Modal-Epistemic Arithmetic (MEA), a theory introduced by Horsten to interpret Epistemic Arithmetic (EA), which in turn was introduced by Shapiro to interpret Heyting Arithmetic. I will show how to interpret MEA in EA such that one can prove that the interpretation of EA is MEA is faithful. Moreover, I will show that one can get rid of a particular Platonist assumption. Then I will discuss models for MEA in light of (...) the problems of logical omniscience and logical competence. Awareness models, impossible worlds models and syntactical models have been introduced to deal with the first problem. Certain conditions on the accessibility relations are needed to deal with the second problem. I go on to argue that those models are subject to the problem of quantifying in, for which I will provide a solution. (shrink)

Given a classical theory T, a Kripke model K for the language L of T is called T-normal or locally PA just in case the classical L-structure attached to each node of K is a classical model of T. Van Dalen, Mulder, Krabbe, and Visser showed that Kripke models of Heyting Arithmetic (HA) over finite frames are locally PA, and that Kripke models of HA over frames ordered like the natural numbers contain infinitely many PA-nodes. We show that (...) Kripke models of the latter sort are in fact PA-normal. This result is extended to a somewhat larger class of frames. (shrink)

This paper proves some independence results for weak fragments of Heyting arithmetic by using Kripke models. We present a necessary condition for linear Kripke models of arithmetical theories which are closed under the negative translation and use it to show that the union of the worlds in any linear Kripke model of HA satisfies PA. We construct a two-node PA-normal Kripke structure which does not force iΣ2. We prove i∀1 ⊬ i∃1, i∃1 ⊬ i∀1, iΠ2 ⊬ iΣ2 and (...) iΣ2 ⊬ iΠ2. We use Smorynski's operation Σ′ to show HA ⊬ lΠ1. (shrink)

The Gödel–Artemov framework offered a formalization of the Brouwer–Heyting–Kolmogorov semantics of intuitionistic logic via classical proofs. In this framework, the intuitionistic propositional logic IPC is embedded in the modal logic S4, S4 is realized in the Logic of Proofs LP, and LP has a provability interpretation in Peano Arithmetic. Self-referential LP-formulas of the type ‘t is a proof of a formula ϕ containing t itself’ are permitted in the realization of S4 in LP, and if such formulas are (...) indeed involved, it is then necessary to use fixed-point arithmetical methods to explain intuitionistic logic via provability. The natural question of whether self-referentiality can be avoided in realization of S4 was answered negatively by Kuznets who provided an S4-theorem that cannot be realized without using directly self-referential LP-formulas. This paper studies the question of whether IPC can be embedded in S4 and then realized in LP without using self-referential formulas. We consider a general class of Gödel-style modal embeddings of IPC in S4 and by extending Kuznetsʼ method, show that there are IPC-theorems such that, under each such embedding, are mapped to S4-theorems that cannot be realized in LP without using directly self-referential formulas. Interestingly, all double-negations of tautologies that are not IPC-theorems, like ¬¬, are shown to require direct self-referentiality. Another example is found in IPC→, the purely implicational fragment of IPC. This suggests that the BHK semantics of intuitionistic logic is intrinsically self-referential. (shrink)

In the first section of this paper we show that i Π1 ≡ W⌝⌝lΠ1 and that a Kripke model which decides bounded formulas forces iΠ1 if and only if the union of the worlds in any path in it satisflies IΠ1. In particular, the union of the worlds in any path of a Kripke model of HA models IΠ1. In the second section of the paper, we show that for equivalence of forcing and satisfaction of Πm-formulas in a linear Kripke (...) model deciding Δ0-formulas it is necessary and sufficient that the model be Σm-elementary. This implies that if a linear Kripke model forces PEMprenex, then it forces PEM. We also show that, for each n ≥ 1, iΦn does not prove ℋ(IΠn's are Burr's fragments of HA. (shrink)

This paper is concerned with the logical structure of arithmetical theories. We survey results concerning logics and admissible rules of constructive arithmetical theories. We prove a new theorem: the admissible propositional rules of Heyting Arithmetic are the same as the admissible propositional rules of Intuitionistic Propositional Logic. We provide some further insights concerning predicate logical admissible rules for arithmetical theories.

The aim of this paper is to formulate and study two weak axiom systems for the conceptual framework of constructive set theory . Arithmetical CST is just strong enough to represent the class of von Neumann natural numbers and its arithmetic so as to interpret Heyting Arithmetic. Rudimentary CST is a very weak subsystem that is just strong enough to represent a constructive version of Jensenʼs rudimentary set theoretic functions and their theory. The paper is a contribution (...) to the study of formal systems for CST that capture significant stages in the development of constructive mathematics in CST. (shrink)

A new method of "minimal" realizability is proposed and applied to show that the definable functions of Heyting arithmetic (HA)--functions f such that HA $\vdash \forall x\exists!yA(x, y)\Rightarrow$ for all m, A(m, f(m)) is true, where A(x, y) may be an arbitrary formula of L(HA) with only x, y free--are precisely the provably recursive functions of the classical Peano arithmetic (PA), i.e., the $ -recursive functions. It is proved that, for prenex sentences provable in HA, Skolem functions (...) may always be chosen to be $ -recursive. The method is extended to intuitionistic finite-type arithmetic, HA ω 0 , and elementary analysis. Generalized forms of Kreisel's characterization of the provably recursive functions of PA and of the no-counterexample-interpretation for PA are consequently derived. (shrink)

We refine the arithmetical hierarchy of various classical principles by finely investigating the derivability relations between these principles over Heyting arithmetic. We mainly investigate some restricted versions of the law of excluded middle, De Morgan's law, the double negation elimination, the collection principle and the constant domain axiom.

We give a corrected proof of the main result in the paper [2] mentioned in the title.

We construct ω-framed Kripke models of i∀1 and iΠ1 non of whose worlds satisfies ∀x∃y(x=2y∨x=2y+1) and ∀x,y∃zExp(x, y, z) respectively. This will enable us to show that i∀1 does not prove ¬¬∀x∃y(x=2y∨x=2y+1) and iΠ1 does not prove ¬¬∀x, y∃zExp(x, y, z). Therefore, i∀1⊬¬¬lop and iΠ1⊬¬¬iΣ1. We also prove that HA⊬lΣ1 and present some remarks about iΠ2.

L.E.J. Brouwer famously took the subject’s intuition of time to be foundational and from there ventured to build up mathematics. Despite being largely critical of formal methods, Brouwer valued axiomatic systems for their use in both communication and memory. Through the Dutch Mathematical Society, Gerrit Mannoury posed a challenge in 1927 to provide an axiomatization of intuitionistic arithmetic. Arend Heyting’s 1928 axiomatization was chosen as the winner and has since enjoyed the status of being the de facto formalization (...) of intuitionistic arithmetic. We argue that axiomatizations of intuitionistic arithmetic ought to make explicit the role of the subject’s activity in the intuitionistic arithmetical process. While Heyting Arithmetic is useful when we want to contrast constructed objects with platonistic ones, Heyting Arithmetic omits the contribution of the subject and thus falls short as a response to Mannoury’s challenge. We offer our own solution, Doxastic Heyting Arithmetic, or $${\textsf{DHA}}$$, which we contend axiomatizes Brouwerian intuitionistic arithmetic. (shrink)

At the beginning of the twentieth century, among the foundational schools of mathematics appeared ‘intuitionism’ by Dutchman L. E. J. Brouwer, who based arithmetic on the intuition of time and all mental constructions that could be made out of it. His pupil Arend Heyting was the first populariser of intuitionism, and he repeatedly emphasised that no philosophy was required to practise intuitionism so that such mathematics could be shared by anyone. Still, stimulated by invitations to humanistic conferences, he (...) wrote a series of notes, preserved in the State Archives, Haarlem, about solipsism. In them, he operated a series of theoretical reflections consisting of the stripping away of the patterns that are part of our consciousness and their subsequent progressive re-introduction, in order to understand the formation of our Self as distinct from the natural world and other humans. In 1996, following a stroke, neuroscientist Jill Bolte Taylor experienced the deprivation of certain abilities in her brain and their successive regaining. In her experience there are remarkable similarities with what Heyting had hypothesised about the formation of the self. The purpose of this article is to highlight them and point out how Heyting's theoretical construct was found to be embodied in the brain. (shrink)

This paper is concerned with notions of consequence. On the one hand, we study admissible consequence, specifically for substitutions of Σ 1 0 -sentences over Heyting arithmetic . 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 Σ 1 0 -substitutions over HA coincides with NNIL -preservativity over intuitionistic propositional (...) logic . Here NNIL is the class of propositional formulas with no nestings of implications to the left . The identical embedding of IPC -derivability into a consequence relation 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. (shrink)

In this paper, we show that the predicate logics of consistent extensions of Heyting's Arithmetic plus Church's Thesis with uniqueness condition are complete $\Pi _{2}^{0}$. Similarly, we show that the predicate logic of HA*, i.e. Heyting's Arithmetic plus the Completeness Principle (for HA*) is complete $\Pi _{2}^{0}$. These results extend the known results due to Valery Plisko. To prove the results we adapt Plisko's method to use Tennenbaum's Theorem to prove 'categoricity of interpretations' under certain assumptions.

We introduce a herbrandized functional interpretation of a first-order semi-intuitionistic extension of Heyting Arithmetic and study its main properties. We then extend the interpretation to a certain system of second-order arithmetic which includes a (classically false) formulation of the FAN principle and weak König's lemma. It is shown that any first-order formula provable in this system is classically true. It is perhaps worthy of note that, in our interpretation, second-order variables are interpreted by finite sets of natural (...) numbers. (shrink)

Using a slight generalization, due to Palmgren, of sheaf semantics, we present a term-model construction that assigns a model to any first-order intuitionistic theory. A modification of this construction then assigns a nonstandard model to any theory of arithmetic, enabling us to reproduce conservation results of Moerdijk and Palmgren for nonstandard Heyting arithmetic. Internalizing the construction allows us to strengthen these results with additional transfer rules; we then show that even trivial transfer axioms or minor strengthenings of (...) these rules destroy conservativity over HA. The analysis also shows that nonstandard HA has neither the disjunction property nor the explicit definability property. Finally, careful attention to the complexity of our definitions allows us to show that a certain weak fragment of intuitionistic nonstandard arithmetic is conservative over primitive recursive arithmetic. (shrink)

Using a slight generalization, due to Palmgren, of sheaf semantics, we present a term-model construction that assigns a model to any first-order intuitionistic theory. A modification of this construction then assigns a nonstandard model to any theory of arithmetic, enabling us to reproduce conservation results of Moerdijk and Palmgren for nonstandard Heyting arithmetic. Internalizing the construction allows us to strengthen these results with additional transfer rules; we then show that even trivial transfer axioms or minor strengthenings of (...) these rules destroy conservativity over HA. The analysis also shows that nonstandard HA has neither the disjunction property nor the explicit definability property. Finally, careful attention to the complexity of our definitions allows us to show that a certain weak fragment of intuitionistic nonstandard arithmetic is conservative over primitive recursive arithmetic. (shrink)

We give a constructive analysis of learning as it arises in various computational interpretations of classical Peano Arithmetic, such as Aschieri and Berardi learning based realizability, Avigad’s update procedures and epsilon substitution method. In particular, we show how to compute in Gödel’s system T upper bounds on the length of learning processes, which are themselves represented in T through learning based realizability. The result is achieved by the introduction of a new non standard model of Gödel’s T, whose new (...) basic objects are pairs of non standard natural numbers and moduli of convergence, where the latter are objects giving constructive information about the former. As a foundational corollary, we obtain that that learning based realizability is a constructive interpretation of Heyting Arithmetic plus excluded middle over Σ10 formulas and of all Peano Arithmetic when combined with Gödel’s double negation translation. As a byproduct of our approach, we also obtain a new proof of Avigad’s theorem for update procedures and thus of the termination of the epsilon substitution method for PA. (shrink)

This paper presents a novel proof of the conservativity of the intuitionistic theory of strictly positive fixpoints, $$\widehat{{\textrm{ID}}}{}_{1}^{{\textrm{i}}}{}$$ ID ^ 1 i, over Heyting arithmetic ($${\textrm{HA}}$$ HA ), originally proved in full generality by Arai (Ann Pure Appl Log 162:807–815, 2011. https://doi.org/10.1016/j.apal.2011.03.002). The proof embeds $$\widehat{{\textrm{ID}}}{}_{1}^{{\textrm{i}}}{}$$ ID ^ 1 i into the corresponding theory over Beeson’s logic of partial terms and then uses two consecutive interpretations, a realizability interpretation of this theory into the subtheory generated by almost negative (...) fixpoints, and a direct interpretation into Heyting arithmetic with partial terms using a hierarchy of satisfaction predicates for almost negative formulae. It concludes by applying van den Berg and van Slooten’s result (Indag Math 29:260–275, 2018. https://doi.org/10.1016/j.indag.2017.07.009) that Heyting arithmetic with partial terms plus the schema of self realizability for arithmetic formulae is conservative over $${\textrm{HA}}$$ HA. (shrink)

Based on the paper [4] we show that true second‐order arithmetic is interpretable over the real‐algebraic structure of models of intuitionistic analysis built upon a certain class of complete Heyting algebras.

We define a class of first-order formulas $\mathsf{P}^{\ast }$ which exactly contains formulas $\varphi$ such that satisfaction of $\varphi$ in any classical structure attached to a node of a Kripke model of intuitionistic predicate logic deciding atomic formulas implies its forcing in that node. We also define a class of $\mathsf{E}$-formulas with the property that their forcing coincides with their classical satisfiability in Kripke models which decide atomic formulas. We also prove that any formula with this property is an $\mathsf{E}$-formula. (...) Kripke models of intuitionistic arithmetical theories usually have this property. As a consequence, we prove a new conservativity result for Peano arithmetic over Heyting arithmetic. (shrink)

The purpose is to study the strength of Ramsey’s Theorem for pairs restricted to recursive assignments ofk-many colors, with respect to Intuitionistic Heyting Arithmetic. We prove that for every natural number$k \ge 2$, Ramsey’s Theorem for pairs and recursive assignments ofkcolors is equivalent to the Limited Lesser Principle of Omniscience for${\rm{\Sigma }}_3^0$formulas over Heyting Arithmetic. Alternatively, the same theorem over intuitionistic arithmetic is equivalent to: for every recursively enumerable infinitek-ary tree there is some$i < k$and (...) some branch with infinitely many children of indexi. (shrink)

This paper is concerned with notions of consequence. On the one hand, we study admissible consequence, specifically for substitutions of Σ 1 0 -sentences over Heyting arithmetic . 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 Σ 1 0 -substitutions over HA coincides with NNIL -preservativity over intuitionistic propositional (...) logic . Here NNIL is the class of propositional formulas with no nestings of implications to the left . The identical embedding of IPC -derivability into a consequence relation 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. (shrink)