Independence results for weak systems of intuitionistic arithmetic

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.     

Generic cuts in models of arithmetic

We present some general results concerning the topological space of cuts of a countable model of arithmetic given by a particular indicator Y. The notion of “indicator” is de.ned in a novel way, without initially specifying what property is indicated and is used to de.ne a topological space of cuts of the model. Various familiar properties of cuts (strength, regularity, saturation, coding properties) are investigated in this sense, and several results are given stating whether or not the set of cuts having the property is comeagre. A new notion of “generic cut” is introduced and investigated and it is shown in the case of countable arithmetically saturated models M ? PA that generic cuts exist, indeed the set of generic cuts is comeagre in the sense of Baire, and furthermore that two generic cuts within the same “small interval” of the model are conjugate by an automorphism of the model.The paper concludes by outlining some applications to constructions of cuts satisfying properties incompatible with genericity, and discussing in model‐theoretic terms those properties for which there is an indicator Y. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Expanding the additive reduct of a model of Peano arithmetic

Let M be a model of first order Peano arithmetic ( PA ) and I an initial segment of M that is closed under multiplication. LetM₀ be the {0, 1,+}‐reduct ofM. We show that there is another model N of PA that is also an expansion of M₀ such that a · ^M a = a · ^N a if and only if a ∈ I for all a ∈ M.     

Intuitionistic weak arithmetic

We construct -framed Kripke models of i₁ and i₁ non of whose worlds satisfies xy(x=2yx=2y+1) and x,yzExp(x, y, z) respectively. This will enable us to show that i₁ does not prove ¬¬xy(x=2yx=2y+1) and i₁ does not prove ¬¬x, yzExp(x, y, z). Therefore, i₁¬¬lop and i₁¬¬i₁. We also prove that HAl₁ and present some remarks about i₂. Mathematics Subject Classification (2000):03F30, 03F55, 03H15.     

Consistency of Heyting arithmetic in natural deduction

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 (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On two questions about feasibly constructive arithmetic

IPV is the intuitionistic theory axiomatized by Cook's equational theory PV plus PIND on NP‐formulas. Two extensions of IPV were introduced by Buss and by Cook and Urquhart by adding PIND for formulas of the form A(x) ∨ B, respectively ¬¬A(x), where A(x) is NP and x is not free in B. Cook and Urquhart posed the question of whether these extensions are proper. We show that in each of the two cases the extension is proper unless the polynomial hierarchy collapses.     

Submodels of Kripke models

A Kripke model ? is a submodel of another Kripke model ℳ if ? is obtained by restricting the set of nodes of ℳ. In this paper we show that the class of formulas of Intuitionistic Predicate Logic that is preserved under taking submodels of Kripke models is precisely the class of semipositive formulas. This result is an analogue of the Łoś-Tarski theorem for the Classical Predicate Calculus. In Appendix A we prove that for theories with decidable identity we can take as the embeddings between domains in Kripke models of the theory, the identical embeddings. This is a well known fact, but we know of no correct proof in the literature. In Appendix B we answer, negatively, a question posed by Sam Buss: whether there is a classical theory T, such that ℋT is HA. Here ℋT is the theory of all Kripke models ℳ such that the structures assigned to the nodes of ℳ all satisfy T in the sense of classical model theory. Received: 4 February 1999 / Published online: 25 January 2001     

The shortest definition of a number in Peano arithmetic

The shortest definition of a number by a first order formula with one free variable, where the notion of a formula defining a number extends a notion used by Boolos in a proof of the Incompleteness Theorem, is shown to be non computable. This is followed by an examination of the complexity of sets associated with this function.     

The arithmetic of cuts in models of arithmetic

We present a number of results on the structure of initial segments of models of Peano arithmetic with the arithmetic operations of addition, subtraction, multiplication, division, exponentiation and logarithm. Each of the binary operations introduced is defined in two dual ways, often with quite different results, and we attempt to systematise the issues and show how various calculations may be carried out. To understand the behaviour of addition and subtraction we introduce a notion of derivative on cuts, analogous to differentiation in the calculus. Multiplication, division and other operations are described by higher order versions of derivative. The work here is presented as important preliminary work related to a nonstandard measure theory of non‐definable bounded subsets of a model of Peano arithmetic.     

Ordinal notations and well-orderings in bounded arithmetic

This paper investigates provability and non-provability of well-foundedness of ordinal notations in weak theories of bounded arithmetic. We define a notion of well-foundedness on bounded domains. We show that T₂¹ and S₂² can prove the well-foundedness on bounded domains of the ordinal notations below ₀ and Γ₀. As a corollary, the class of polynomial local search problems, PLS, can be augmented with cost functions that take ordinal values below ₀ and Γ₀ without increasing the class PLS.     

Weak Arithmetics and Kripke Models

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 (complete) 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 Π_m‐formulas in a linear Kripke model deciding Δ₀‐formulas it is necessary and sufficient that the model be Σ_m‐elementary. This implies that if a linear Kripke model forces PEM_prenex, then it forces PEM. We also show that, for each n ≥ 1, iΦ_n does not prove ℋ︁(IΠ_n's are Burr's fragments of HA.     

Substructure lattices and almost minimal end extensions of models of Peano arithmetic

This paper concerns intermediate structure lattices Lt(??/??), where ?? is an almost minimal elementary end extension of the model ?? of Peano Arithmetic. For the purposes of this abstract only, let us say that ?? attains L if L ? Lt(??/??) for some almost minimal elementary end extension of ??. If T is a completion of PA and L is a finite lattice, then: (A) If some model of T attains L, then every countable model of T does. (B) If some rather classless, ?₁‐saturated model of T attains L, then every model of T does. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Corrigendum to “Weak Arithmetics and Kripke Models”

We give a corrected proof of the main result in the paper [2] mentioned in the title. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the local distribution of certain arithmetic functions

Let d(n), σ ₁(n), and φ(n) stand for the number of positive divisors of n, the sum of the positive divisors of n, and Euler’s function, respectively. For each ν ∈, Z, we obtain asymptotic formulas for the number of integers n ⩽ x for which e _n = 2 ^v r for some odd integer m as well as for the number of integers n ⩽ x for which e _n = 2 ^v r for some odd rational number r. Our method also applies when φ(n) is replaced by σ ₁(n), thus, improving upon an earlier result of Bateman, Erdős, Pomerance, and Straus, according to which the set of integers n such that is an integer is of density 1/2. Research supported in part by a grant from NSERC. Research supported by the Applied Number Theory Research Group of the Hungarian Academy of Science and by a grant from OTKA. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 3, pp. 315–331, July–September, 2006.     

On the structure of initial segments of models of arithmetic

For any countable nonstandard modelM of a sufficiently strong fragment of arithmeticT, and any nonstandard numbersa, c M, Mca, there is a modelK ofT which agrees withM up toa and such that inK there is a proof of contradiction inT with Gödel number .     

On a variance associated with the distribution of primes in arithmetic progressions

As is usual in prime number theory, write It is well known that when q is close to x the averagevalue of is about xlog q,and recently Friedlander and Goldston have shown that if then the first moment of V(x,q)-U(x,q)is small. In this memoir it is shown that the same is true forall moments. 2000 Mathematics Subject Classification: 11N13.     

Nonstandard models that are definable in models of Peano Arithmetic

In this paper, we investigate definable models of Peano Arithmetic PA in a model of PA. For any definable model N without parameters in a model M, we show that N is isomorphic to M if M is elementary extension of the standard model and N is elementarily equivalent to M. On the other hand, we show that there is a model M and a definable model N with parameters in M such that N is elementarily equivalent to M but N is not isomorphic to M. We also show that there is a model M and a definable model N with parameters in M such that N is elementarily equivalent to M, and N is isomorphic to M, but N is not definably isomorphic to M. And also, we give a generalization of Tennenbaum's theorem. At the end, we give a new method to construct a definable model by a refinement of Kotlarski's method. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)