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 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).

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).

Inconsistency of uncountable infinite sets under ZFC framework

Kybernetes ◽  
2008 ◽  
Vol 37 (3/4) ◽  
pp. 453-457 ◽  
Author(s):  
Wujia Zhu ◽  
Yi Lin ◽  
Guoping Du ◽  
Ningsheng Gong
Keyword(s):  
Natural Number ◽  
Set Theory ◽  
Design Methodology ◽  
Natural Numbers ◽  
Real Numbers ◽  
Conceptual Approach ◽  
Content Type ◽  
Infinite Sets ◽  
Axiomatic Set Theory ◽  
First Time

PurposeThe purpose is to show that all uncountable infinite sets are self‐contradictory non‐sets.Design/methodology/approachA conceptual approach is taken in the paper.FindingsGiven the fact that the set N={x|n(x)} of all natural numbers, where n(x)=df “x is a natural number” is a self‐contradicting non‐set in this paper, the authors prove that in the framework of modern axiomatic set theory ZFC, various uncountable infinite sets are either non‐existent or self‐contradicting non‐sets. Therefore, it can be astonishingly concluded that in both the naive set theory or the modern axiomatic set theory, if any of the actual infinite sets exists, it must be a self‐contradicting non‐set.Originality/valueThe first time in history, it is shown that such convenient notion as the set of all real numbers needs to be reconsidered.

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 Divisibility Networks of the Rational Numbers in the Unit Interval

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1879
Author(s):  
Pedro A. Solares-Hernández ◽  
Miguel A. García-March ◽  
J. Alberto Conejero
Keyword(s):  
Real Life ◽  
Rational Numbers ◽  
Prime Numbers ◽  
Unit Interval ◽  
Natural Numbers ◽  
Scale Free ◽  
Free Distribution ◽  
Human Interventions ◽  
Diagonal Argument ◽  
Infinite Sets

Divisibility networks of natural numbers present a scale-free distribution as many other process in real life due to human interventions. This was quite unexpected since it is hard to find patterns concerning anything related with prime numbers. However, it is by now unclear if this behavior can also be found in other networks of mathematical nature. Even more, it was yet unknown if such patterns are present in other divisibility networks. We study networks of rational numbers in the unit interval where the edges are defined via the divisibility relation. Since we are dealing with infinite sets, we need to define an increasing covering of subnetworks. This requires an order of the numbers different from the canonical one. Therefore, we propose the construction of four different orders of the rational numbers in the unit interval inspired in Cantor’s diagonal argument. We motivate why these orders are chosen and we compare the topologies of the corresponding divisibility networks showing that all of them have a free-scale distribution. We also discuss which of the four networks should be more suitable for these analyses.

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”).

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 On the set of finite subsets of a set

1975 ◽  
Vol 20 (1) ◽  
pp. 38-45
Author(s):  
J. L Hickman
Keyword(s):  
Natural Numbers ◽  
Finite Sets ◽  
Infinite Sets

We sometimes think of medial (that is, infinite Dedekind-finite) sets as being “small” infinite sets. Medial cardinals can be defined as those cardinals that are incomparable to ℵℴ; hence we tend to think of them as being spread out on a plane “just above” the natural numbers, which seems to lend support to the view expressed above that medial sets are “small”.

scholarly journals On a Sumset Conjecture of Erdős

2015 ◽  
Vol 67 (4) ◽  
pp. 795-809 ◽  
Author(s):  
Mauro Di Nasso ◽  
Isaac Goldbring ◽  
Renling Jin ◽  
Steven Leth ◽  
Martino Lupini ◽  
...  
Keyword(s):  
Positive Solution ◽  
Asymptotic Density ◽  
Natural Numbers ◽  
Amenable Groups ◽  
Banach Density ◽  
Infinite Sets ◽  
Erdös Conjecture

AbstractErdős conjectured that for any set A ⊆ ℕ with positive lower asymptotic density, there are infinite sets B;C ⊆ ℕ such that B + C ⊆ A. We verify Erdős’ conjecture in the case where A has Banach density exceeding ½ . As a consequence, we prove that, for A ⊆ ℕ with positive Banach density (amuch weaker assumption than positive lower density), we can find infinite B;C ⊆ ℕ such that B+C is contained in the union of A and a translate of A. Both of the aforementioned results are generalized to arbitrary countable amenable groups. We also provide a positive solution to Erdős’ conjecture for subsets of the natural numbers that are pseudorandom.

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