Measuring the size of infinite collections of natural numbers: Was Cantor’s theory of infinite number inevitable?

2016 ◽  
pp. 116-153
Author(s):  
Paolo Mancosu
Keyword(s):  
Infinite Number ◽  
Natural Numbers

Diophantine quadruples and near-Diophantine quintuples from P3,K sequences

2017 ◽  
Vol 10 (01) ◽  
pp. 1750010
Author(s):  
A. M. S. Ramasamy
Keyword(s):  
Infinite Number ◽  
Fibonacci Numbers ◽  
Natural Numbers ◽  
Text Type ◽  
Type Formula ◽  
Fourth Term ◽  
Unexplored Area ◽  
Diophantine Quintuples ◽  
Diophantine Quadruples

The question of a non-[Formula: see text]-type [Formula: see text] sequence wherein the fourth term shares the property [Formula: see text] with the first term has not been investigated so far. The present paper seeks to fill up the gap in this unexplored area. Let [Formula: see text] denote the set of all natural numbers and [Formula: see text] the sequence of Fibonacci numbers. Choose two integers [Formula: see text] and [Formula: see text] with [Formula: see text] such that their product increased by [Formula: see text] is a square [Formula: see text]. Certain properties of the sequence [Formula: see text] defined by the relation [Formula: see text] are established in this paper and polynomial expressions for Diophantine quadruples from the [Formula: see text] sequence [Formula: see text] are derived. The concept of a near-Diophantine quintuple is introduced and it is proved that there exist an infinite number of near-Diophantine quintuples.

scholarly journals The Methods of Solving Equations Ax+By=Cz with Co-Prime A, B, C, where x≥2,y≥2,x≥2 Are Natural Numbers, Equal the Two only in one of the Three Possible Cases - The Proof of Catalan's Conjecture

2013 ◽  
Vol 8 ◽  
pp. 17-25
Author(s):  
K. Raja Rama Gandhi ◽  
Reuven Tint ◽  
Michael Tint
Keyword(s):  
Infinite Number ◽  
Natural Numbers ◽  
Whole Numbers ◽  
Solving Equations ◽  
Solutions Of Equations

One of the principal problems of the Beal's conjecture, as we see that, is methods for finding a pairwise coprime solution which is defined below. First found methods and identities, allowing receiving infinite number solutions of equations as Ax+By=Cz for co-prime integers arranged in a pair (A,B,C)=1 are natural (whole) numbers, where a fixed permutation (x,y,z)corresponds to each of the permutations (2,3,4), (2,4,3), (4,3,2) Here we obtain also our method and identities of all not recurrent and not co-prime solutions of the above type, part of which has already been published, in contrast to the method of obtaining the recurrence not co-prime solutions of this type from [(1), W. Sierpiński, p. 21-25, 63]. As the solution of the main problem appeared additional problems that solved by obtained appropriate identities. Given as two equal proofs of Catalan's Conjecture.

Non-constructive rules of inference

2018 ◽  
Author(s):  
A.P. Hazen
Keyword(s):  
Natural Number ◽  
Infinite Number ◽  
Deductive System ◽  
Higher Order ◽  
The Other ◽  
Natural Numbers ◽  
Substitutional Quantification ◽  
Deductive Systems ◽  
Formal Systems ◽  
Order Arithmetic

For some theoretical purposes, generalized deductive systems (or, ‘semi-formal’ systems) are considered, having rules with an infinite number of premises. The best-known of these rules is the ‘ω-rule’, or rule of infinite induction. This rule allows the inference of ∀nΦ(n) from the infinitely many premises Φ(0), Φ(1),… that result from replacing the numerical variable n in Φ(n) with the numeral for each natural number. About 1930, in part as a response to Gödel’s demonstration that no formal deductive system had as theorems all and only the true formulas of arithmetic, several writers (most notably, Carnap) suggested considering the semi-formal systems obtained, from some formulation of arithmetic, by adding this rule. Since no finite notation can provide terms for all sets of natural numbers, no comparable rule can be formulated for higher-order arithmetic. In effect, the ω-rule is valid just in case the relevant quantifier can be interpreted substitutionally; looked at from the other side, the validity of some analogue of the ω-rule is the essential mathematical characteristic of substitutional quantification.

DISTRIBUTION OF PRIME AND COMPOSITE NUMBERS AND THEIR ALGORHYTHMIC APPENDICES

2017 ◽  
pp. 48-64 ◽  
Author(s):  
Sergey Ivanovich Chermidov
Keyword(s):  
Infinite Number ◽  
Diophantine Equations ◽  
Prime Numbers ◽  
Natural Numbers ◽  
Ordinal Numbers ◽  
Independent Verification ◽  
Composite Number ◽  
Verification Test

The article focuses on methods defining and distributing the composite numbers, prime numbers, twins of prime numbers and composite numbers of twins that do not have divisors 2 and 3 in N , based on the set of numbers of type Θ = {6 k ± 1 / kN } where N is the set of all natural numbers, which is a semigroup with respect to multiplication. The calculation the exact quantity of primes in a given interval is given. A method for obtaining prime numbers p ≥ 5 by their ordinal numbers in a set of primes p ≥ 5 is proposed, as well as a new algorithm for finding and distributing prime numbers on the basis of the closeness of the set Θ. The article shows that any composite number n Θ is representable as products (6 x ± 1) (6 y ± 1), where x, yN are the natural solutions of one of the four Diophantine equations P ( x , y , λ) = 0 : 6 × xy ± x ± y - λ = 0. It has been proved that if there is a parameter λ of twins of prime numbers, then none of the Diophantine P ( x , y , λ) = 0 equations has any solutions. A new universal, deterministic, polynomial and independent verification test is provided for the simplicity of the numbers of a species 6 × k ± 1. Algorithms of distributions of parameters of twins of prime numbers and parameters composite numbers of twins are given, they are not divisible by 2 and 3, and variants of proofs for their infinite number are given.

scholarly journals MEASURING THE SIZE OF INFINITE COLLECTIONS OF NATURAL NUMBERS: WAS CANTOR’S THEORY OF INFINITE NUMBER INEVITABLE?

2009 ◽  
Vol 2 (4) ◽  
pp. 612-646 ◽  
Author(s):  
PAOLO MANCOSU
Keyword(s):  
Infinite Number ◽  
Philosophy Of Mathematics ◽  
Natural Numbers ◽  
Finite Sets ◽  
New Developments ◽  
Rational Nature ◽  
Cardinal Numbers ◽  
History Of ◽  
Infinite Sets ◽  
One To One

Cantor’s theory of cardinal numbers offers a way to generalize arithmetic from finite sets to infinite sets using the notion of one-to-one association between two sets. As is well known, all countable infinite sets have the same ‘size’ in this account, namely that of the cardinality of the natural numbers. However, throughout the history of reflections on infinity another powerful intuition has played a major role: if a collection A is properly included in a collection B then the ‘size’ of A should be less than the ‘size’ of B (part–whole principle). This second intuition was not developed mathematically in a satisfactory way until quite recently. In this article I begin by reviewing the contributions of some thinkers who argued in favor of the assignment of different sizes to infinite collections of natural numbers (Thabit ibn Qurra, Grosseteste, Maignan, Bolzano). Then, I review some recent mathematical developments that generalize the part–whole principle to infinite sets in a coherent fashion (Katz, Benci, Di Nasso, Forti). Finally, I show how these new developments are important for a proper evaluation of a number of positions in philosophy of mathematics which argue either for the inevitability of the Cantorian notion of infinite number (Gödel) or for the rational nature of the Cantorian generalization as opposed to that, based on the part–whole principle, envisaged by Bolzano (Kitcher).

scholarly journals Los infinitos de algunas series divergentes

2020 ◽  
Vol 20 (2) ◽  
Author(s):  
Diego Miramontes de León ◽  
Gerardo Miramontes de León
Keyword(s):  
Infinite Number ◽  
The Other ◽  
Divergent Series ◽  
Harmonic Series ◽  
Natural Numbers

En este trabajo interesa mostrar que dos series divergentes, aunque ambas tienen un número infinito de términos, al tener términos diferentes, su valor al infinito también difiere. En el documento se muestra que la serie armónica, dada por la suma del inverso de los números naturales, puede descomponerse en dos series. Una de ellas dada por la suma del inverso de los naturales de la forma 1/np con p > 1 y la otra, que será llamada subarmónica, formada por el resto de los términos que completan la serie armónica original. Se muestra que cada una de estas series es, una convergente y la otra divergente, obteniendo así la serie original divergente. Se incluye la demostración de la divergencia de las nuevas series, y como extensión de esta descomposición de la serie armónica, se hace una comparación de dos series subarmónicas las cuales, a pesar de ser ambas divergentes, difieren en su valor al infinito.   Abstract This work aims to show that two divergent series, although both have an infinite number of terms, if they have different terms, their value to infinity also differs. In this document, it is shown that the harmonic series, given by the sum of the inverse of natural numbers, can be decomposed into two series; one of them is given by the sum of the inverse of the naturals in the form 1/np where p > 1 and the other, which will be called subharmonic, formed by the rest of the terms that complete the originalharmonic series. It is shown that each of these series is one convergent and the other divergent, thus obtaining the original divergent series. It is included the demonstration of the divergence of the new series, and as an extension of this decomposition of the harmonic series, a comparison is made of two subharmonic series which, despite being both divergent, differ in their value to infinity.

What is the purpose of cognition?

2020 ◽  
Vol 43 ◽  
Author(s):  
Aba Szollosi ◽  
Ben R. Newell
Keyword(s):  
Infinite Number ◽  
Human Cognition ◽  
The Way

Abstract The purpose of human cognition depends on the problem people try to solve. Defining the purpose is difficult, because people seem capable of representing problems in an infinite number of ways. The way in which the function of cognition develops needs to be central to our theories.

The Problem of Alternative Monotheisms: Another Serious Challenge to Theism

2018 ◽  
Vol 10 (1) ◽  
pp. 31-51
Author(s):  
Raphael Lataster
Keyword(s):  
Infinite Number ◽  
Current Evidence ◽  
God Concept ◽  
Classical Theism ◽  
The World ◽  
Divine Transcendence ◽  
Hiddenness Of God

Theistic and analytic philosophers of religion typically privilege classical theism by ignoring or underestimating the great threat of alternative monotheisms.[1] In this article we discuss numerous god-models, such as those involving weak, stupid, evil, morally indifferent, and non-revelatory gods. We find that theistic philosophers have not successfully eliminated these and other possibilities, or argued for their relative improbability. In fact, based on current evidence – especially concerning the hiddenness of God and the gratuitous evils in the world – many of these hypotheses appear to be more probable than theism. Also considering the – arguably infinite – number of alternative monotheisms, the inescapable conclusion is that theism is a very improbable god-concept, even when it is assumed that one and only one transcendent god exists.[1] I take ‘theism’ to mean ‘classical theism’, which is but one of many possible monotheisms. Avoiding much of the discussion around classical theism, I wish to focus on the challenges in arguing for theism over monotheistic alternatives. I consider theism and alternative monotheisms as entailing the notion of divine transcendence.

On Ramsey Minimal Graphs

10.37236/1184 ◽  
1994 ◽  
Vol 1 (1) ◽  
Author(s):  
Tomasz Łuczak
Keyword(s):  
Infinite Number ◽  
Probabilistic Argument ◽  
Minimal Graphs

An elementary probabilistic argument is presented which shows that for every forest $F$ other than a matching, and every graph $G$ containing a cycle, there exists an infinite number of graphs $J$ such that $J\to (F,G)$ but if we delete from $J$ any edge $e$ the graph $J-e$ obtained in this way does not have this property.

Sign in / Sign up

Export Citation Format

Share Document