Decidability and Undecidability Results on the Modal μ-Calculus with a Natural Number-Valued Semantics

2010 ◽  
pp. 148-160
Author(s):  
Alexis Goyet ◽  
Masami Hagiya ◽  
Yoshinori Tanabe
Keyword(s):  
Natural Number

The Natural Numbers

2018 ◽  
Author(s):  
Øystein Linnebo
Keyword(s):  
Natural Number ◽  
Peano Arithmetic ◽  
Number Sequence ◽  
Natural Numbers ◽  
Ordinal Numbers ◽  
Cardinal Numbers

How are the natural numbers individuated? That is, what is our most basic way of singling out a natural number for reference in language or in thought? According to Frege and many of his followers, the natural numbers are cardinal numbers, individuated by the cardinalities of the collections that they number. Another answer regards the natural numbers as ordinal numbers, individuated by their positions in the natural number sequence. Some reasons to favor the second answer are presented. This answer is therefore developed in more detail, involving a form of abstraction on numerals. Based on this answer, a justification for the axioms of Dedekind–Peano arithmetic is developed.

scholarly journals The Limits of Computation

Axiomathes ◽  
2021 ◽  
Author(s):  
Andrew Powell
Keyword(s):  
Natural Number ◽  
Set Theory ◽  
Lambda Calculus ◽  
Second Order ◽  
Functional Interpretation ◽  
Arbitrary Type ◽  
Computable Functions

AbstractThis article provides a survey of key papers that characterise computable functions, but also provides some novel insights as follows. It is argued that the power of algorithms is at least as strong as functions that can be proved to be totally computable in type-theoretic translations of subsystems of second-order Zermelo Fraenkel set theory. Moreover, it is claimed that typed systems of the lambda calculus give rise naturally to a functional interpretation of rich systems of types and to a hierarchy of ordinal recursive functionals of arbitrary type that can be reduced by substitution to natural number functions.

scholarly journals ADDITIVE BASES AND NIVEN NUMBERS

2021 ◽  
pp. 1-8
Author(s):  
CARLO SANNA
Keyword(s):  
Positive Integer ◽  
Natural Number ◽  
Riemann Hypothesis ◽  
Additive Basis

Abstract Let $g \geq 2$ be an integer. A natural number is said to be a base-g Niven number if it is divisible by the sum of its base-g digits. Assuming Hooley’s Riemann hypothesis, we prove that the set of base-g Niven numbers is an additive basis, that is, there exists a positive integer $C_g$ such that every natural number is the sum of at most $C_g$ base-g Niven numbers.

A simple proof of existence of weak and statistical solutions of Navier–Stokes equations

1992 ◽  
Vol 436 (1896) ◽  
pp. 1-11 ◽  
Keyword(s):  
Weak Solution ◽  
Initial Data ◽  
Natural Number ◽  
Simple Proof ◽  
Stokes Equations ◽  
Galerkin Approximation ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Existence Proofs ◽  
Statistical Solutions

The Galerkin approximation to the Navier–Stokes equations in dimension N , where N is an infinite non-standard natural number, is shown to have standard part that is a weak solution. This construction is uniform with respect to non-standard representation of the initial data, and provides easy existence proofs for statistical solutions.

scholarly journals Band description of knots and Vassiliev invariants

2002 ◽  
Vol 133 (2) ◽  
pp. 325-343 ◽  
Author(s):  
KOUKI TANIYAMA ◽  
AKIRA YASUHARA
Keyword(s):  
Natural Number ◽  
Algebraic Condition ◽  
Vassiliev Invariants

In the 1990s, Habiro defined Ck-move of oriented links for each natural number k [5]. A Ck-move is a kind of local move of oriented links, and two oriented knots have the same Vassiliev invariants of order [les ] k−1 if and only if they are transformed into each other by Ck-moves. Thus he has succeeded in deducing a geometric conclusion from an algebraic condition. However, this theorem appears only in his recent paper [6], in which he develops his original clasper theory and obtains the theorem as a consequence of clasper theory. We note that the ‘if’ part of the theorem is also shown in [4], [9], [10] and [16], and in [13] Stanford gives another characterization of knots with the same Vassiliev invariants of order [les ] k−1.

Theory of models with generalized atomic formulas

1960 ◽  
Vol 25 (1) ◽  
pp. 1-26 ◽  
Author(s):  
H. Jerome Keisler
Keyword(s):  
Natural Number ◽  
Finite Sequence ◽  
Sufficient Condition ◽  
Elementary Extension ◽  
Necessary And Sufficient Condition ◽  
Elementary Class ◽  
Relational Systems ◽  
Necessary And Sufficient

IntroductionWe shall prove the following theorem, which gives a necessary and sufficient condition for an elementary class to be characterized by a set of sentences having a prescribed number of alternations of quantifiers. A finite sequence of relational systems is said to be a sandwich of order n if each is an elementary extension of (i ≦ n—2), and each is an extension of (i ≦ n—2). If K is an elementary class, then the statements (i) and (ii) are equivalent for each fixed natural number n.

scholarly journals A GENERALISED KUMMER'S CONJECTURE

2010 ◽  
Vol 52 (3) ◽  
pp. 453-472 ◽  
Author(s):  
M. J. R. MYERS
Keyword(s):  
Natural Number ◽  
Class Numbers ◽  
Cyclotomic Fields ◽  
Rate Of Growth ◽  
Relative Class ◽  
Almost All

AbstractKummer's conjecture predicts the rate of growth of the relative class numbers of cyclotomic fields of prime conductor. We extend Kummer's conjecture to cyclotomic fields of conductor n, where n is any natural number. We show that the Elliott–Halberstam conjecture implies that this generalised Kummer's conjecture is true for almost all n but is false for infinitely many n.

What is a natural number?

1988 ◽  
Vol 22 (2) ◽  
pp. 103-113 ◽  
Author(s):  
Noel Balzer
Keyword(s):  
Natural Number

Complete compositions of a natural number

2006 ◽  
Vol 29 (2) ◽  
pp. 229-238
Author(s):  
Richard Warlimont
Keyword(s):  
Natural Number

GENERALIZING THE TITCHMARSH DIVISOR PROBLEM

2012 ◽  
Vol 08 (03) ◽  
pp. 613-629 ◽  
Author(s):  
ADAM TYLER FELIX
Keyword(s):  
Natural Number ◽  
Asymptotic Relation ◽  
Divisor Problem ◽  
Implied Constant ◽  
Artin's Conjecture ◽  
Primitive Roots

Let a be a natural number different from 0. In 1963, Linnik proved the following unconditional result about the Titchmarsh divisor problem [Formula: see text] where c is a constant dependent on a. Titchmarsh proved the above result assuming GRH for Dirichlet L-functions in 1931. We establish the following asymptotic relation: [Formula: see text] where Ck is a constant dependent on k and a, and the implied constant is dependent on k. We also apply it a question related to Artin's conjecture for primitive roots.

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