Intuitionistic weak arithmetic (original) (raw)

Independence results for weak systems of intuitionistic arithmetic

MLQ, 2003

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 P A. We construct a two-node P A-normal Kripke structure which does not force iΣ 2. We prove i∀ 1 i∃ 1 , i∃ 1 i∀ 1 , iΠ 2 iΣ 2 and iΣ 2 iΠ 2. We use Smorynski's operation Σ to show HA lΠ 1 .

Weak Arithmetics and Kripke Models

MLQ, 2002

In the first section of this paper we show that iΠ 1 ≡ W ¬¬lΠ 1. In the second section of the paper, we show that for equivalence of forcing and satisfaction of Π m-formulas in a linear Kripke model deciding ∆ 0-formulas, it is necessary and sufficient that the model be Σ m-elementary. This implies that if a linear Kripke model forces P EM prenex , then it forces P EM. We also show that, for each n 1, iΦ n does not prove H(IΠ n). Here, Φ n 's are Burr's fragments of HA.

Classical and Intuitionistic Models of Arithmetic

Notre Dame Journal of Formal Logic 37, 1996, 452–461, 1996

Given a classical theory T, a Kripke structure K is called T-normal (or locally T) if for each node of K, the structure attached to it is a classical model of T. It has been known for some time now, thanks to van Dalen, Mulder, Krabbe, and Visser, that Kripke models of HA over finite frames are locally PA. They also proved that models of HA over the frame (ω,≤) contain infinitely many Peano nodes. We will show that such models are in fact PA-normal, that is, they consist entirely of Peano nodes. These results are then applied to a somewhat larger class of frames. We close with some general considerations on properties of non-Peano nodes in arbitrary models of HA.

An Independence Result for Intuitionistic Bounded Arithmetic

Journal of Logic and Computation, 2006

It is shown that the intuitionistic theory of polynomial induction on positive Π b 1 (coNP) formulas does not prove the sentence ¬¬∀x, y∃z ≤ y(x ≤ |y| → x = |z|). This implies the unprovability of the scheme ¬¬PIND(Σ b+ 1) in the mentioned theory. However, this theory contains the sentence ∀x, y¬¬∃z ≤ y(x ≤ |y| → x = |z|). The above independence result is proved by constructing an ω-chain of submodels of a countable model of S 2 + Ω 3 + ¬exp such that none of the worlds in the chain satisfies the sentence, and interpreting the chain as a Kripke model.

Model theory of bounded arithmetic with applications to independence results

Logic in Tehran, 2000

In this paper we apply some new and some old methods in order to construct classical and intuitionistic models for theories of bounded arithmetic. We use these models to obtain proof theoretic consequences. In particular, we construct an ωchain of models of BASIC such that the union of its worlds satisfies S 1 2 but none of its worlds satisfies the sentence ∀x∃y(x = 0 ∨ x = y + 1). Interpreting this chain as a Kripke model shows that double negation of the above mentioned sentence is not provable in the intuitionistic theory of BASIC plus polynomial induction on coNP formulas.

Inconsistent Models (and Infinite Models) for Arithmetics with Constructible Falsity

Logic and Logical Philosophy, 2019

An earlier paper on formulating arithmetic in a connexive logic ended with a conjecture concerning C♯ , the closure of the Peano axioms in Wansing's connexive logic C. Namely, the paper conjectured that C ♯ is Post consistent relative to Heyting arithmetic, i.e., is nontrivial if Heyting arithmetic is nontrivial. The present paper borrows techniques from relevant logic to demonstrate that C♯ is Post consistent simpliciter, rendering the earlier conjecture redundant. Given the close relationship between C and Nelson's paraconsistent N4, this also supplements Nelson's own proof of the Post consistency of N4♯. Insofar as the present technique allows infinite models, this resolves Nelson's concern that N4 ♯ is of interest only to those accepting that there are finitely many natural numbers.

Intuitionistic axiomatizations for bounded extension Kripke models

Annals of Pure and Applied Logic, 2003

We present axiom systems, and provide soundness and strong completeness theorems, for classes of Kripke models with restricted extension rules among the node structures of the model. As examples we present an axiom system for the class of coÿnal extension Kripke models, and an axiom system for the class of end-extension Kripke models. We also show that Heyting arithmetic (HA) is strongly complete for its class of end-extension models. Coÿnal extension models of HA are models of Peano arithmetic (PA).

On the Hierarchy of Intuitionistic Bounded Arithmetic

Journal of Logic and Computation, 2007

In this paper we are concerned with cuts in models of Samuel Buss' theories of bounded arithmetic, i.e. theories like S i 2 and T i 2. In correspondence with polynomial induction, we consider a rather new notion of cut that we call p-cut. We also consider small cuts, i.e. cuts that are bounded above by a small element. We study the basic properties of pcuts and small cuts. In particular, we prove some overspill and underspill properties for them.

Some Weak Fragments of HA and Certain Closure Properties

Journal of Symbolic Logic, 2002

We show that Intuitionistic Open Induction iop is not closed under the rule DN S(∃ − 1). This is established by constructing a Kripke model of iop+¬L y (2y > x), where L y (2y > x) is universally quantified on x. On the other hand, we prove that iop is equivalent with the intuitionistic theory axiomatized by P A − plus the scheme of weak ¬¬LN P for open formulas, where universal quantification on the parameters precedes double negation. We also show that for any open formula ϕ(y) having only y free, (P A −) i L y ϕ(y). We observe that the theories iop, i∀ 1 and iΠ 1 are closed under Friedman's translation by negated formulas and so under V R and IP. We include some remarks on the classical worlds in Kripke models of iop. 1.1 Let DOR (resp. P A −) be the finite set of usual axioms (including Trichotomy) for discretely ordered commutative rings with 1 (resp. their nonnegative parts) in the language L = {+, ·, < , 0, 1} of arithmetic. Peano Arithmetic P A (resp. Heyting Arithmetic HA) is the classical (resp. intuitionistic, obtained by dropping the principle P EM of excluded middle whose instance P EM ϕ on a formula ϕ is ϕ ∨ ¬ϕ) first order theory axiomatized by P A − together with the induction scheme whose instance with respect to a distinguished free variable x on a formula ϕ(x, y) is I x ϕ = I x ϕ(x, y) : ∀y(ϕ(0, y) ∧ ∀x(ϕ(x, y) → ϕ(x + 1, y)) → ∀xϕ(x, y)).