Decidability and undecidability of theories with a predicate for the primes

1993 ◽  
Vol 58 (2) ◽  
pp. 672-687 ◽  
Author(s):  
P. T. Bateman ◽  
C. G. Jockusch ◽  
A. R. Woods
Keyword(s):  
General Result ◽  
Order Theory ◽  
Second Order ◽  
The Other ◽  
Linear Case ◽  
Natural Numbers ◽  
Successor Function ◽  
First Order ◽  
First Order Theory

AbstractIt is shown, assuming the linear case of Schinzel's Hypothesis, that the first-order theory of the structure 〈ω; +, P〉, where P is the set of primes, is undecidable and, in fact, that multiplication of natural numbers is first-order definable in this structure. In the other direction, it is shown, from the same hypothesis, that the monadic second-order theory of 〈ω S, P〉 is decidable, where S is the successor function. The latter result is proved using a general result of A. L. Semënov on decidability of monadic theories, and a proof of Semënov's result is presented.

More on an undecidability result of Bateman, Jockusch and Woods

1998 ◽  
Vol 63 (1) ◽  
pp. 50-50 ◽  
Author(s):  
M. Boffa
Keyword(s):  
Prime Numbers ◽  
Order Theory ◽  
Linear Case ◽  
Natural Numbers ◽  
Fixed Element ◽  
First Order ◽  
First Order Theory ◽  
Image Position ◽  
Dirichlet Theorem ◽  
Undecidability Result

Let P be the set of prime numbers. Theorem 1 of [1] shows that the linear case of Schinzel's Hypothesis (H) implies that multiplication is definable in 〈ω,+,P〉 and therefore that the first-order theory of this structure is undecidable. Let m be any fixed natural number >2, let R be the set of natural numbers <m which are prime to m, and let r be any fixed element of R. The setis infinite (Dirichlet). Theorem 1 of [1] can be improved as follows:Proposition. The linear case of Schinzel's Hypothesis (H) implies that multiplication is definable in 〈ω,+,Pm,r〉 and therefore that the first-order theory of this structure is undecidable.Proof. We follow [1] with the following new ingredients. Let k be the number of elements of R, i.e. k = ϕ(m) where ϕ is Euler's totient function. Since k is even, the polynomial g(n) = nk + n satisfies g(0) = g(−1) = 0, so (by Lemma 1 of [1]) it follows from the linear case of (H) that there are natural numbers al (l ϵ ω) such that al+g(0), al+g(1),…, al+g(l) are consecutive primes. Since R is finite, we may assume that all the al's have the same residue t in R, so that al+g(i) ≡ t+1+i (mod m) for i ϵ R. This implies that the function t+1+i (reduced mod m) gives a permutation of R, so we can find s ϵ R such that al+g(s) ≡ r (mod m). Consider the polynomial h(n) = g(s + mn) and let bl = as+ml. Then bl + h(0), bl + h(1),…, bl + h(l) are elements of Pm,r. They are not necessarily consecutive elements of Pm,r, but they are separated by a fixed number of elements of Pm,r. This implies that {h(n) ∣ n ϵ ω} is definable in 〈ω,+,Pm,r〉(by adapting the proof of Theorem 1 of [1]), and the result follows.

Degrees of classes of RE sets

1976 ◽  
Vol 41 (3) ◽  
pp. 695-696 ◽  
Author(s):  
J. R. Shoenfield
Keyword(s):  
Order Theory ◽  
Turing Degree ◽  
Natural Numbers ◽  
First Order ◽  
First Order Theory ◽  
Simple Sets ◽  
Natural Classes

In [3], Martin computed the degrees of certain classes of RE sets. To state the results succinctly, some notation is useful.If A is a set (of natural numbers), dg(A) is the (Turing) degree of A. If A is a class of sets, dg(A) = {dg(A): A ∈ A). Let M be the class of maximal sets, HHS the class of hyperhypersimple sets, and DS the class of dense simple sets. Martin showed that dg(M), dg(HHS), and dg(DS) are all equal to the set H of RE degrees a such that a′ = 0″.Let M* be the class of coinfinite RE sets having no superset in M; and define HHS* and DS* similarly. Martin showed that dg(DS*) = H. In [2], Lachlan showed (among other things) that dg(M*)⊆K, where K is the set of RE degrees a such that a″ > 0″. We will show that K ⊆ dg (HHS*). Since maximal sets are hyperhypersimple, this gives dg(M*) = dg (HHS*) = K.These results suggest a problem. In each case in which dg(A) has been calculated, the set of nonzero degrees in dg(A) is either H or K or the empty set or the set of all nonzero RE degrees. Is this always the case for natural classes A? Natural here might mean that A is invariant under all automorphisms of the lattice of RE sets; or that A is definable in the first-order theory of that lattice; or anything else which seems reasonable.

An Experimental Study of Deviations from the Linear Transfer Function

1974 ◽  
Vol 32 ◽  
pp. 400-401
Author(s):  
William A. Voter ◽  
Harold P. Erickson
Keyword(s):  
Experimental Study ◽  
Image Formation ◽  
Order Theory ◽  
Second Order ◽  
Diffraction Spot ◽  
Order Effects ◽  
First Order ◽  
First Order Theory ◽  
Second Order Effects ◽  
The Fourier Transform

In a previous experimental study of image formation using a thin (20 nm) negatively stained catalase crystal, it was found that a linear or first order theory of image formation would explain almost entirely the changes in the Fourier transform of the image as a function of defocus. In this case it was concluded that the image is a valid picture of the object density. For thicker, higher contrast objects the first order theory may not be valid. Second order effects could generate false diffraction spots which would lead to spurious and artifactual image details. These second order effects would appear as deviations of the diffraction spot amplitudes from the first order theory. Small deviations were in fact noted in the study of the thin crystals, but there was insufficient data for a quantitative analysis.

Numerical Calculation of Second-Order Wave Resistance

1977 ◽  
Vol 21 (02) ◽  
pp. 94-106
Author(s):  
Young S. Hong
Keyword(s):  
Numerical Calculation ◽  
Steady Motion ◽  
Order Theory ◽  
Iteration Scheme ◽  
Wave Resistance ◽  
Second Order ◽  
Lagrangian Coordinates ◽  
Restricted Range ◽  
First Order ◽  
First Order Theory

The wave resistance due to the steady motion of a ship was formulated in Lagrangian coordinates by Wehausen [1].2 By introduction of an iteration scheme the solutions for the first order and second order3 were obtained. The draft/length ratio was assumed small in order to simplify numerical computation. In this work Wehausen's formulas are used to compute the resistance numerically. A few models are selected and the wave resistance is calculated. These results are compared with other methods and experiments. Generally speaking, the second-order resistance shows better agreement with experiment than first-order theory in only a restricted range of Froude number, say 0.25 to 0.35, and even here not uniformly. For larger Froude numbers it underestimates seriously.

Polynomial rings and weak second-order logic

1985 ◽  
Vol 50 (4) ◽  
pp. 953-972 ◽  
Author(s):  
Anne Bauval
Keyword(s):  
Zero Divisor ◽  
Order Theory ◽  
Second Order ◽  
Order Logic ◽  
Polynomial Rings ◽  
Commutative Field ◽  
First Order ◽  
First Order Theory ◽  
Order Language ◽  
Second Order Logic

This article is a rewriting of my Ph.D. Thesis, supervised by Professor G. Sabbagh, and incorporates a suggestion from Professor B. Poizat. My main result can be crudely summarized (but see below for detailed statements) by the equality: first-order theory of F[Xi]i∈I = weak second-order theory of F.§I.1. Conventions. The letter F will always denote a commutative field, and I a nonempty set. A field or a ring (A; +, ·) will often be written A for short. We shall use symbols which are definable in all our models, and in the structure of natural numbers (N; +, ·):— the constant 0, defined by the formula Z(x): ∀y (x + y = y);— the constant 1, defined by the formula U(x): ∀y (x · y = y);— the operation ∹ x − y = z ↔ x = y + z;— the relation of division: x ∣ y ↔ ∃ z(x · z = y).A domain is a commutative ring with unity and without any zero divisor.By “… → …” we mean “… is definable in …, uniformly in any model M of L”.All our constructions will be uniform, unless otherwise mentioned.§I.2. Weak second-order models and languages. First of all, we have to define the models Pf(M), Sf(M), Sf′(M) and HF(M) associated to a model M = {A; ℐ) of a first-order language L [CK, pp. 18–20]. Let L1 be the extension of L obtained by adjunction of a second list of variables (denoted by capital letters), and of a membership symbol ∈. Pf(M) is the model (A, Pf(A); ℐ, ∈) of L1, (where Pf(A) is the set of finite subsets of A. Let L2 be the extension of L obtained by adjunction of a second list of variables, a membership symbol ∈, and a concatenation symbol ◠.

Complex language, complex thought?

2016 ◽  
Vol 33 ◽  
pp. 28-40
Author(s):  
Suzanne T.M. Bogaerds-Hazenberg ◽  
Petra Hendriks
Keyword(s):  
Theory Of Mind ◽  
Order Theory ◽  
Second Order ◽  
False Belief ◽  
Production Task ◽  
False Belief Task ◽  
First Order ◽  
First Order Theory ◽  
Complex Thought

Abstract It has been argued (e.g., by De Villiers and colleagues) that the acquisition of sentence embedding is necessary for the development of first-order Theory of Mind (ToM): the ability to attribute beliefs to others. This raises the question whether the acquisition of double embedded sentences is related to, and perhaps even necessary for, the development of second-order ToM: the ability to attribute beliefs about beliefs to others. This study tested 55 children (aged 7-10) on their ToM understanding in a false-belief task and on their elicited production of sentence embeddings. We found that second-order ToM passers produced mainly double embeddings, whereas first-order ToM passers produced mainly single embeddings. Furthermore, a better performance on second-order ToM predicted a higher rate of double embeddings and a lower rate of single embeddings in the production task. We conclude that children’s ability to produce double embeddings is related to their development of second-order ToM.

Application des miroirs électrostatiques à l'élimination de l'effet chromatique dans les spectromètres magnétiques. Première partie: théorie linéaire

1969 ◽  
Vol 47 (3) ◽  
pp. 331-340 ◽  
Author(s):  
Marcel Baril
Keyword(s):  
Matrix Algebra ◽  
Order Term ◽  
Order Theory ◽  
Second Order ◽  
Energy Dispersive ◽  
Powerful Method ◽  
First Order ◽  
First Order Theory ◽  
Electrostatic Mirror ◽  
Premiere Partie

Combining an energy-dispersive element with a magnetic prism results in an achromatic mass dispersive instrument, if parameters are chosen appropriately. A plane electrostatic mirror has been chosen as the energy-dispersive element. Trajectories are described in terms of lateral, angular, and energy variations about the principal trajectory. Achromatism and conjugate plane conditions have been calculated by the powerful method of matrix algebra. The first order theory is given in this article (part one), the second order term will be studied in part two which will be published later.

Categoricity and the natural numbers

2018 ◽  
Author(s):  
Tim Button ◽  
Sean Walsh
Keyword(s):  
Order Theory ◽  
Second Order ◽  
Order Logic ◽  
Natural Numbers ◽  
Categorical Theory ◽  
First Order Theory ◽  
Standard Models ◽  
Skolem Theorem ◽  
Infinite Model ◽  
Second Order Logic

This chapter focuses on modelists who want to pin down the isomorphism type of the natural numbers. This aim immediately runs into two technical barriers: the Compactness Theorem and the Löwenheim-Skolem Theorem (the latter is proven in the appendix to this chapter). These results show that no first-order theory with an infinite model can be categorical; all such theories have non-standard models. Other logics, such as second-order logic with its full semantics, are not so expressively limited. Indeed, Dedekind's Categoricity Theorem tells us that all full models of the Peano axioms are isomorphic. However, it is a subtle philosophical question, whether one is entitled to invoke the full semantics for second-order logic — there are at least four distinct attitudes which one can adopt to these categoricity result — but moderate modelists are unable to invoke the full semantics, or indeed any other logic with a categorical theory of arithmetic.

LOGICAL ASPECTS OF CAYLEY-GRAPHS: THE MONOID CASE

2006 ◽  
Vol 16 (02) ◽  
pp. 307-340 ◽  
Author(s):  
DIETRICH KUSKE ◽  
MARKUS LOHREY
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Order Theory ◽  
Second Order ◽  
Point Of View ◽  
Direct Products ◽  
Graph Products ◽  
Arbitrary Graph ◽  
First Order ◽  
First Order Theory

Cayley-graphs of monoids are investigated under a logical point of view. It is shown that the class of monoids, for which the Cayley-graph has a decidable monadic second-order theory, is closed under free products. This result is derived from a result of Walukiewicz, stating that the decidability of monadic second-order theories is preserved under tree-like unfoldings. Concerning first-order logic, it is shown that the class of monoids, for which the Cayley-graph has a decidable first-order theory, is closed under arbitrary graph products, which generalize both, free and direct products. For the proof of this result, tree-like unfoldings are generalized to so-called factorized unfoldings. It is shown that the decidability of the first-order theory of a structure is preserved by factorized unfoldings. Several additional results concerning factorized unfoldings are shown.

A NEW FORMALIZATION OF FEFERMAN’S SYSTEM OF FUNCTIONS AND CLASSES AND ITS RELATION TO FREGE STRUCTURE

1995 ◽  
Vol 06 (03) ◽  
pp. 187-202 ◽  
Author(s):  
SUSUMU HAYASHI ◽  
SATOSHI KOBAYASHI
Keyword(s):  
Finite Number ◽  
Order Theory ◽  
Point Of View ◽  
The Other ◽  
Technical Result ◽  
Technical Point ◽  
First Order ◽  
First Order Theory

A new axiomatization of Feferman’s systems of functions and classes1,2 is given. The new axiomatization has a finite number of class constructors resembling the proposition constructors of Frege structure by Aczel.3 Aczel wrote “It appears that from the technical point of view the two approaches (Feferman’s system and Frege structure) run parallel to each other in the sense that any technical result for one approach can be reconstructed for the other”.3 By the aid of the new axiomatization, Aczel’s observation becomes so evident. It is now straightforward to give a mutual interpretation between our formulation and a first order theory of Frege structure, which improve results by Beeson in Ref. 4.

Sign in / Sign up

Export Citation Format

Share Document