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.

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).

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 $\mathbb N$.}

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).

Doughnuts, floating ordinals, square brackets, and ultraflitters

2000 ◽  
Vol 65 (1) ◽  
pp. 461-473 ◽  
Author(s):  
Carlos A. Di Prisco ◽  
James M. Henle
Keyword(s):  
Axiom Of Choice ◽  
Inaccessible Cardinal ◽  
Natural Numbers ◽  
Real Numbers ◽  
Power Set ◽  
Infinite Sets ◽  
The Axiom Of Choice ◽  
Infinite Set ◽  
Infinite Sequences

In this paper, we study partition properties of the set of real numbers. The meaning of “set of real numbers” will vary, referring at times to the collection of sequences of natural numbers, ωω; the collection of infinite sets of natural numbers [ω]ω; the collection of infinite sequences of zeroes and ones, 2ω; or (ω), the power set of ω.The archetype for the relations is the property: “all sets of reals are Ramsey,” in the notation of Erdős and Hajnal, ω → (ω)ω. This states that for every partition F : [ω]ω → 2, there is an infinite set H ∈ [ω]ω such that F is constant on [H]ω. Like virtually all of the properties we will discuss, it contradicts the Axiom of Choice but is compatible with the principle of dependent choices (DC). DC will be used throughout the paper wihtout further mention.The properties discussed in this paper will vary in two respects. Some, like ω → (ω)ω, will be incompatible with the existence of an ultrafilter on ω (UF) and some will not. Some are known to be consistent relative to ZF alone, and for some, such as ω → (ω)ω, the question is still open. All properties, however, are true in Solovay's model and hence are consistent relative to Con(ZF + “there exists an inaccessible cardinal”).

scholarly journals A New Set Theory for Analysis

2019 ◽  
Author(s):  
Juan P. 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 $\mathbb N$.}

scholarly journals On radicals of infinite matrix rings

1969 ◽  
Vol 16 (3) ◽  
pp. 195-203 ◽  
Author(s):  
A. D. Sands
Keyword(s):  
Infinite Matrix ◽  
Natural Numbers ◽  
Index Set ◽  
Matrix Rings ◽  
Infinite Set

Let R be a ring and I an infinite set. We denote by M(R) the ring of row finite matrices over I with entries in R. The set I will be omitted from the notation, as the same index set will be used throughout the paper. For convenience it will be assumed that the set of natural numbers is a subset of I.

A continuous generalization of the transversal property

1984 ◽  
Vol 95 (1) ◽  
pp. 21-23
Author(s):  
P. Komjáth
Keyword(s):  
Finite Subset ◽  
Measure Space ◽  
Choice Function ◽  
Finite Measure ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finite Sets ◽  
Real Numbers ◽  
One To One ◽  
Necessary And Sufficient

A transversal for a set-system is a one-to-one choice function. A necessary and sufficient condition for the existence of a transversal in the case of finite sets was given by P. Hall (see [4, 3]). The corresponding condition for the case when countably many countable sets are given was conjectured by Nash-Williams and later proved by Damerell and Milner [2]. B. Bollobás and N. Varopoulos stated and proved the following measure theoretic counterpart of Hall's theorem: if (X, μ) is an atomless measure space, ℋ = {Hi: i∈I} is a family of measurable sets with finite measure, λi (i∈I) are non-negative real numbers, then we can choose a subset Ti ⊆ Hi with μ(Ti) = λi and μ(Ti ∩ Ti′) = 0 (i ≠ i′) if and only if μ({U Hi: iεJ}) ≥ Σ{λi: iεJ}: for every finite subset J of I. In this note we generalize this result giving a necessary and sufficient condition for the case when I is countable and X is the union of countably many sets of finite measure.

scholarly journals SOME GROUP THEORETIC ASPECTS OF t-NORMS

2000 ◽  
Vol 08 (01) ◽  
pp. 1-6 ◽  
Author(s):  
FRED RICHMAN ◽  
ELBERT WALKER
Keyword(s):  
Automorphism Group ◽  
Multiplicative Group ◽  
Order Relation ◽  
Unit Interval ◽  
Real Numbers ◽  
Positive Real ◽  
Explicit Formulas ◽  
One To One ◽  
The Right ◽  
Usual Order

Let A be the automorphism group of the unit interval with its usual order relation, and let ℝ+ be the embedding of the multiplicative group of positive real numbers into A given by exponentiation. Strict t-norms are in one-to-one correspondence with the right cosets of ℝ+ in A. Here, we identify the normalizer of ℝ+ in A and give explicit formulas for the corresponding set of t-norms.

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).

Sign in / Sign up

Export Citation Format

Share Document