scholarly journals The Diagonalization Paradox - Expanded

2020 ◽  
Author(s):  
Ron Ragusa
Keyword(s):  
Initial Conditions ◽  
Georg Cantor ◽  
Natural Numbers ◽  
Real Numbers ◽  
The Real ◽  
One To One ◽  
The One

In 1891 Georg Cantor published his Diagonal Method which, he asserted, proved that the real numbers cannot be put into a one-to-one correspondence with the natural numbers. In this paper we will see how by varying the initial conditions of Cantor’s proof we can use the diagonal method to produce a one-to-one correspondence between the set of natural numbers and the set of infinite binary decimals in the interval (0, 1). In the appendix we demonstrate that using the diagonal method recursively will, at the limit of the process, fully account for all the infinite binary decimals in (0, 1). The proof will cement the one-to-one correspondence between the natural numbers and the infinite binary decimals in (0, 1).

scholarly journals The Diagonalization Paradox

2020 ◽  
Author(s):  
Ron Ragusa
Keyword(s):  
Initial Conditions ◽  
Open Interval ◽  
Georg Cantor ◽  
Natural Numbers ◽  
Real Numbers ◽  
The Real ◽  
Diagonal Argument ◽  
One To One

In 1891 Georg Cantor published his Diagonal Argument which, he asserted, proved that the real numbers cannot be put into a one-to-one correspondence with the natural numbers. In this paper we will see how by varying the initial conditions of the demonstration we can use Cantor’s method to produce a one-to-one correspondence between the set of natural numbers and the set of infinite binary decimals in the open interval (0, 1).

Proceed with care, because some infinities are larger than others

2020 ◽  
pp. 281-292
Author(s):  
Susan D'Agostino
Keyword(s):  
Number Line ◽  
Mathematics Students ◽  
Natural Numbers ◽  
Real Numbers ◽  
Countable Infinity ◽  
One To One ◽  
Infinite Set

“Proceed with care, because some infinities are larger than others” explains in detail why the infinite set of real numbers—all of the numbers on the number line—represents a far larger infinity than the infinite set of natural numbers—the counting numbers. Readers learn to distinguish between countable infinity and uncountable infinity by way of a method known as a “one-to-one correspondence.” Mathematics students and enthusiasts are encouraged to proceed with care in both mathematics and life, lest they confuse countable infinity with uncountable infinity, large with unfathomably large, or order with disorder. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

An Editor Recalls Some Hopeless Papers

1998 ◽  
Vol 4 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Wilfrid Hodges
Keyword(s):  
Natural Numbers ◽  
Real Numbers ◽  
Diagonal Argument ◽  
The One ◽  
Academy Of Sciences ◽  
Main Message

§1. Introduction. I dedicate this essay to the two-dozen-odd people whose refutations of Cantor's diagonal argument (I mean the one proving that the set of real numbers and the set of natural numbers have different cardinalities) have come to me either as referee or as editor in the last twenty years or so. Sadly these submissions were all quite unpublishable; I sent them back with what I hope were helpful comments. A few years ago it occurred to me to wonder why so many people devote so much energy to refuting this harmless little argument—what had it done to make them angry with it? So I started to keep notes of these papers, in the hope that some pattern would emerge.These pages report the results. They might be useful for editors faced with similar problem papers, or even for the authors of the papers themselves. But the main message to reach me is that there are several points of basic elementary logic that we usually teach and explain very badly, or not at all.In 1995 an engineer named William Dilworth, who had published a refutation of Cantor's argument in the Transactions of the Wisconsin Academy of Sciences, Arts and Letters, sued for libel a mathematician named Underwood Dudley who had called him a crank ([9] pp. 44f, 354).

Transdenumerability of the reals

2020 ◽  
Author(s):  
Matheus Pereira Lobo
Keyword(s):  
Real Numbers ◽  
The Real ◽  
One To One

We show that the real numbers are transdenumerable by setting a one to one map with the set of the transfinite ordinals introduced by Cantor.

Countable Compagtifications

1966 ◽  
Vol 18 ◽  
pp. 616-620 ◽  
Author(s):  
Kenneth D. Magill
Keyword(s):  
Positive Integer ◽  
Compact Space ◽  
Real Line ◽  
Topological Spaces ◽  
Dense Subspace ◽  
Natural Numbers ◽  
Finite Sets ◽  
The Real ◽  
One To One

It is assumed that all topological spaces discussed in this paper are Hausdorff. By a compactification αX of a space X we mean a compact space containing X as a dense subspace. If, for some positive integer n, αX — X consists of n points, we refer to αX as an n-point compactification of X, in which case we use the notation αn X. If αX — X is countable, we refer to αX as a countable compactification of X. In this paper, the statement that a set is countable means that its elements are in one-to-one correspondence with the natural numbers. In particular, finite sets are not regarded as being countable. Those spaces with n-point compactifications were characterized in (3). From the results obtained there it followed that the only n-point compactifications of the real line are the well-known 1- and 2-point compactifications and the only n-point compactification of the Euclidean N-space, EN (N > 1), is the 1-point compactification.

Quadratic forms ‘à la’ local theory

1967 ◽  
Vol 63 (3) ◽  
pp. 579-586 ◽  
Author(s):  
A. Fröhlich
Keyword(s):  
Quadratic Forms ◽  
The Other ◽  
Local Theory ◽  
Real Numbers ◽  
The Real ◽  
Other Hand ◽  
Non Local ◽  
Characteristic 2 ◽  
The One ◽  
Adic Case

In this note (cf. sections 3, 4) I shall give an axiomatization of those fields (of characteristic ≠ 2) which have a theory of quadratic forms like the -adic numbers or like the real numbers. This leads then, for instance, to a generalization of the well-known theorems on -adic forms to a wider class of fields, including non-local ones. The main purpose of the exercise is, however, to separate out the roles of the arithmetic in the underlying field, on the one hand, which solely enters into the verification of the axioms, and of the ordinary algebra of quadratic forms on the other hand. The resulting clarification of the structure of the theory is of interest even in the known -adic case.

On virtual classes and real numbers

1950 ◽  
Vol 15 (2) ◽  
pp. 131-134
Author(s):  
R. M. Martin
Keyword(s):  
Standard Procedure ◽  
Homogeneous System ◽  
Alternative Methods ◽  
Natural Numbers ◽  
Real Numbers ◽  
The Real ◽  
Virtual Class ◽  
Virtual Classes ◽  
Main Ideas

In a simple, applied functional calculus of first order (i.e., one admitting no functional variables but at least one functional constant), abstracts or schematic expressions may be introduced to play the role of variables over designatable sets or classes. The entities or quasi-entities designated or quasi-designated by such abstracts may be called, following Quine, virtual classes and relations. The notion of virtual class is always relative to a given formalism and depends upon what functional constants are taken as primitive. The first explicit introduction of a general notation for virtual classes (relative to a given formalism) appears to be D4.1 of the author's A homogeneous system for formal logic. That paper develops a system admitting only individuals as values for variables and is adequate for the theory of general recursive functions of natural numbers. Numbers and functions are in fact identified with certain kinds of virtual classes and relations.In the present paper it will be shown how certain portions of the theory of real numbers can be constructed upon the basis of the theory of virtual classes and relations of H.L.The method of building up the real numbers to be employed is essentially an adaptation of standard procedure. Although the main ideas underlying this method are well known, the mirroring of these ideas within the framework of the restricted concepts admitted here presents possibly some novelty. In particular, a basis for the real numbers is provided which in no way admits classes or relations or other "abstract" objects as values for variables. Presupposing the natural numbers, the essential steps are to construct the simple rationals as virtual dyadic relations between natural numbers, to construct the generalized or signed rationals as virtual tetradic relations among natural numbers, and then to formulate a notation for real numbers as virtual classes (of a certain kind) of generalized rationals. Of course, there are several alternative methods. This procedure, however, appears to correspond more to the usual one.

scholarly journals The way to list all infinite real numbers and to construct a bijection between natural numbers and real numbers

2021 ◽  
Author(s):  
Shee-Ping Chen
Keyword(s):  
Georg Cantor ◽  
Natural Numbers ◽  
Real Numbers ◽  
Diagonal Argument ◽  
Infinite Sets ◽  
The Way

Abstract Georg Cantor defined countable and uncountable sets for infinite sets. The set of natural numbers is defined as a countable set, and the set of real numbers is proved to be uncountable by Cantor’s diagonal argument. Most scholars accept that it is impossible to construct a bijection between the set of natural numbers and the set of real numbers. However, the way to construct a bijection between the set of natural numbers and the set of real numbers is proposed in this paper. The set of real numbers can be proved to be countable by Cantor’s definition. Cantor’s diagonal argument is challenged because it also can prove the set of natural numbers to be uncountable. The process of argumentation provides us new perspectives to consider about the size of infinite sets.

scholarly journals Russian Dolls

1978 ◽  
Vol 21 (2) ◽  
pp. 237-240 ◽  
Author(s):  
J. B. Wilker
Keyword(s):  
Natural Numbers ◽  
Real Numbers ◽  
The Real ◽  
Strengthened Version

In an earlier number of this Bulletin, P. Erdös [1] posed the following problem. “For each line ℓ of the plane, Aℓ is a segment of ℓ. Show that the set ∪ℓAℓ contains the sides of a triangle.” One objective of this paper is to prove a strengthened version of this result in iV-dimensions. As usual denotes the cardinality of the natural numbers and c, the cardinality of the real numbers.

scholarly journals A New Set Theory for Analysis

2019 ◽  
Author(s):  
Juan Ramirez
Keyword(s):  
Real Number ◽  
Number System ◽  
Number Line ◽  
Physical Models ◽  
Order Structure ◽  
Natural Numbers ◽  
Real Numbers ◽  
The Real ◽  
Power Set ◽  
Real Number System

We present the real number system as a natural generalization of the natural numbers. First, we prove the co-finite topology, $Cof(\mathbb N)$, is isomorphic to the natural numbers. Then, we generalize these results to describe the continuum $[0,1]$. Then we prove the power set $2^{\mathbb Z}$ contains a subset isomorphic to the non-negative real numbers, with all its defining structure of operations and order. Finally, we provide two different constructions of the entire real number line. We see that the power set $2^{\mathbb N}$ can be given the defining structure of $\mathbb R$. The constructions here provided give simple rules for calculating addition, multiplication, subtraction, division, powers and rational powers of real numbers, and logarithms. The supremum and infimum are explicitly constructed by means of a well defined algorithm that ends in denumerable steps. In section 5 we give evidence our construction of $\mathbb N$ and $\mathbb R$ are canonical; these constructions are as natural as possible. In the same section, we propose a new axiomatic basis for analysis. In the last section we provide a series of graphic representations and physical models that can be used to represent the real number system. We conclude that the system of real numbers is completely defined by the order structure of $\mathbb N$.}

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