Heyting Arithmetic Research Articles

Substitutions of Σ10-sentences: explorations between intuitionistic propositional logic and intuitionistic arithmetic

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 ( 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 Σ 1 0-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.

Analyzing realizability by Troelstra's methods

Realizabilities are powerful tools for establishing consistency and independence results for theories based on intuitionistic logic. Troelstra discovered principles ECT 0 and GC 1 which precisely characterize formal number and function realizability for intuitionistic arithmetic and analysis, respectively. Building on Troelstra's results and using his methods, we introduce the notions of Church domain and domain of continuity in order to demonstrate the optimality of “almost negativity” in ECT 0 and GC 1; strengthen “double negation shift” DNS 0 to DNS 1 and use it with GC 1 and Markov's Principle MP to characterize the classically provably realizable formulas of analysis; and show that intuitionistic analysis with GC 1, MP and DNS 1 is consistent and satisfies Troelstra's and Church's Rules. We end with a discussion of semi-intuitionistic theories and a conjecture.

Extracting information from intermediate semiconstructive HA-systems – extended abstract

In this abstract we will describe research in progress on the problem of extracting information from proofs. Here we will concentrate our attention on semiconstructive calculi, which is a kind of calculus that is of interest in the framework of program synthesis and formal verification. We will discuss the notion of uniformly semiconstructive calculus, introduce our information extraction mechanism and apply it to two calculi extending Intuitionistic Arithmetic.

Intuitionistic Choice and Restricted Classical Logic

Recently, Coquand and Palmgren considered systems of intuitionistic arithmetic in all finite types together with various forms of the axiom of choice and a numerical omniscience schema (NOS) which implies classical logic for arithmetical formulas. Feferman subsequently observed that the proof theoretic strength of such systems can be determined by functional interpretation based on a non-constructive μ-operator and his well-known results on the strength of this operator from the 70’s. In this note we consider a weaker form LNOS (lesser numerical omniscience schema) of NOS which suffices to derive the strong form of binary Konig’s lemma studied by Coquand/Palmgren and gives rise to a new and mathematically strong semi-classical system which, nevertheless, can proof theoretically be reduced to primitive recursive arithmetic PRA. The proof of this fact relies on functional interpretation and a majorization technique developed in a previous paper. ∗Basic Research in Computer Science, Centre of the Danish National Research Foundation.

Fragments of Heyting arithmetic

AbstractWe define classes Φn of formulae of first-order arithmetic with the following properties:(i) Every φ ϵ Φn is classically equivalent to a Πn-formula (n ≠ 1, Φ1 := Σ1).(ii) (iii) IΠn and iΦn (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 Φn both under existential and universal quantification (we call these classes Θn) the corresponding theories iΘn still prove the same Π2-formulae. In a second part we consider iΔ0 plus collection-principles. We show that both the provably recursive functions and the provably total functions of are polynomially bounded. Furthermore we show that the contrapositive of the collection-schema gives rise to instances of the law of excluded middle and hence .

Classical Arithmetic is Part of Intuitionistic Arithmetic

Classical Arithmetic is Part of Intuitionistic Arithmetic

Note on extensions of Heyting's arithmetic by adding the "creative subject"

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.

A Note on Goodman's Theorem

Goodman's theorem states that intuitionistic arithmetic in all finite types plus full choice, HAω + AC, is conservative over first-order intuitionistic arithmetic HA. We show that this result does not extend to various subsystems of HAω, HA with restricted induction.

Rules and Arithmetics

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

A common axiom set for classical and intuitionistic plane geometry

We describe a first order axiom set which yields the classical first order Euclidean geometry of Tarski when used with classical logic, and yields an intuitionistic (or constructive) Euclidean geometry when used with intuitionistic logic. The first order language has a single six place atomic predicate and no function symbols. The intuitionistic system has a computational interpretation in recursive function theory, that is, a realizability interpretation analogous to those given by Kleene for intuitionistic arithmetic and analysis. This interpretation shows the unprovability in the intuitionistic theory of certain “nonconstructive” theorems of the classical geometry.

Strictly Primitive Recursive Realizability, II. Completeness with Respect to Iterated Reflection and a Primitive Recursive $\omega$-Rule

The notion of strictly primitive recursive realizability is further investigated, and the realizable prenex sentences, which coincide with primitive recursive truths of classical arithmetic, are characterized as precisely those provable in transfinite progressions $\{\mathrm{PRA}(b) \vert b \in \underline{\mathrm{O}}\}$ over a fragment $\mbox{PR-}(\Sigma^{0}_{1}\mbox{-IR})$ of intuitionistic arithmetic. The progressions are based on uniform reflection principles of bounded complexity iterated along initial segments of a primitive recursively formulated system $\mathrm{\underline{O}}$ of notations for constructive ordinals. A semiformal system closed under a primitive recursively restricted $\omega$-rule is described and proved equivalent to the transfinite progressions with respect to the prenex sentences.

Finite Sets and Natural Numbers in Intuitionistic TT Without Extensionality

In this paper, we prove that Heyting's arithmetic can be interpreted in an intuitionistic version of Russell's Simple Theory of Types without extensionality.

Basic Predicate Calculus

We establish a completeness theorem for first-order basic predicate logic BQC, a proper subsystem of intuitionistic predicate logic IQC, using Kripke models with transitive underlying frames. We develop the notion of functional well-formed theory as the right notion of theory over BQC for which strong completeness theorems are possible. We also derive the undecidability of basic arithmetic, the basic logic equivalent of intuitionistic Heyting Arithmetic and classical Peano Arithmetic.

Classical Arithmetic is Part of Intuitionistic Arithmetic

Classical Arithmetic is Part of Intuitionistic Arithmetic

In Defense of Epistemic Arithmetic

This paper presents a defense of Epistemic Arithmetic as used for a formalization of intuitionistic arithmetic and of certain informal mathematical principles. First, objections by Allen Hazen and Craig Smorynski against Epistemic Arithmetic are discussed and found wanting. Second, positive support is given for the research program by showing that Epistemic Arithmetic can give interesting formulations of Church's Thesis.

Minimal models of Heyting arithmetic

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].

Finite Sets and Natural Numbers in Intuitionistic TT

We show how to interpret Heyting's arithmetic in an intuitionistic version of TT, Russell's Simple Theory of Types. We also exhibit properties of finite sets in this theory and compare them with the corresponding properties in classical TT. Finally, we prove that arithmetic can be interpreted in intuitionistic TT$_3$, the subsystem of intuitionistic TT involving only three types. The definitions of intuitionistic TT and its finite sets and natural numbers are obtained in a straightforward way from the classical definitions. This is very natural and seems to make intuitionistic TT an interesting intuitionistic set theory to study, beside intuitionistic ZF.

Undecidability and intuitionistic incompleteness

Let S be a deductive system such that S-derivability (⊢s) is arithmetic and sound with respect to structures of class K. From simple conditions on K and ⊢s, it follows constructively that the K-completeness of ⊢s implies MP(S), a form of Markov's Principle. If ⊢s is undecidable then MP(S) is independent of first-order Heyting arithmetic. Also, if ⊢s is undecidable and the S proof relation is decidable, then MP(S) is independent of second-order Heyting arithmetic, HAS. Lastly, when ⊢s is many-one complete, MP(S) implies the usual Markov's Principle MP.

Realizing Brouwer's sequences

When Kleene extended his recursive realizability interpretation from intuitionistic arithmetic to analysis, he was forced to use more than recursive functions to interpret sequences and conditional constructions. In fact, he used what classically appears to be the full continuum. We describe here a generalization to higher type of Kleene's realizability, one case of which, ( U, R)-realizability, uses general recursive functions throughout, both to realize theorems and to interpret choice sequences. ( U, R)-realizability validates a version of the bar theorem and the usual continuity principles, while also providing naturally, as Kleene's 1965 realizability does not, for versions of lawless sequence axioms, as well as of Church's Thesis.

Logic and computation, Proceedings of a workshop held at Carnegie Mellon University, June 30–July 2, 1987, edited by Wilfried Sieg, Contemporary Mathematics, vol. 106, American Mathematical Society, Providence1990, xiv + 297 pp. - Douglas K. Brown. Notions of closed subsets of a complete separable metric space in weak subsystems of second order arithmetic. Pp. 39–50. - Kostas Hatzikiriakou and Stephen

Logic and computation, Proceedings of a workshop held at Carnegie Mellon University, June 30–July 2, 1987, edited by Wilfried Sieg, Contemporary Mathematics, vol. 106, American Mathematical Society, Providence1990, xiv + 297 pp. - Douglas K. Brown. Notions of closed subsets of a complete separable metric space in weak subsystems of second order arithmetic. Pp. 39–50. - Kostas Hatzikiriakou and Stephen