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INFINITY WITHOUT SIZE a Uniform Conception for All Infinite Sets of Numbers

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INFINITY WITHOUT SIZE A Uniform Conception for All Infinite Sets of Numbers INCLUDES REFUTATIONS OF: CANTOR’S MULTIPLE INFINITIES NIVEN'S ANALYSIS ON MEASURE ZERO AND THE IRRATIONAL NUMBERS THE AXIOM OF CHOOSING AND THE WELL ORDERED THEOREM THE MULTIVERSE THEORY Eli Rapaport Copyright © 2012-2020 All Rights Reserved [email protected] TABLE OF CONTENTS Preface 3 1.0 Countability of Sets 5 1.1 Rational Numbers Countability 5 1.2 Algebraic Numbers Countability 6 2.0 Refutation of Cantor’s Diagonal and Interval Proofs 7 2.1 Cantor’s Diagonal Proof 7 2.2 Refutation of Cantor’s Diagonal Proof 9 2.3 Reinterpreting the List Used for Diagonal Argument 11 2.4 Cantor’s Interval Proof 12 2.5 Refutation of Cantor’s Interval Proof 15 2.6 Comparing the Diagonal and Interval Proofs 15 3.1 Real Numbers are Not Uncountable 15 4.1 Cantor’s Power Set Proof 17 4.2 Refutation of Cantor’s Power Set Proof 18 5.0 A Single Infinity Without Size 21 6.0 Identification of Transcendental Numbers 22 7.1 Niven’s Precepts on Irrational Numbers and Measure Zero 22 7.2 Dismissal of Niven’s Ideas 24 8.1 Flawed Reason for Accepting the Axiom of Choosing 25 8.2 History of the Axiom of Choosing 25 8.3 Rejecting the Axiom of Choosing 27 8.4 Inability to Well Order the Real Numbers 27 8.5 Creating Infinite Cartesian Product Using Axiom of Choosing 29 1 Infinity Without Size 8.6 Infinite Cartesian Product Without Axiom of Choosing 29 9.0 Misguided Reason for Creation of ZF Axiomatic Set Theory 32 9.1 Russell's Paradox 32 9.2 Refutation of Russell's Paradox 32 9.3 Cantor's Paradox 33 9.4 Refutation of Cantor's Paradox 33 10.0 The Multiverse Folly 33 2 Infinity Without Size PREFACE This treatise clarifies the categorization of the various types of infinite sets comprising the real numbers, and considers whether each of these sets are countable. The rational, irrational, algebraic, and transcendental sets of numbers are covered in this text. The term "uncountable set" is used by Cantor to indicate that a particular infinite set has a greater quantity of numbers than the infinite set of natural numbers. However, this concept is meaningless, because infinite sets do not have a size. Therefore, we cannot describe any infinite set as being uncountable in the sense of greater quantity. An infinite set signifies a set of unending potential, not a completed set. The view held by Cantor that there are multiple infinities with different cardinalities is shown to be false. Cantor’s diagonal, interval, and power set proofs are demonstrated as being without basis. Niven’s contention that the set of irrational numbers does not have measure zero is refuted. Cantor's idea of multiple infinities spawned the fictional well ordering theorem, which is accepted as valid for use in mathematical proofs, even though it is based on the mythical axiom of choosing. The inadmissibility of the axiom of choosing and the invalidity of the well ordering theorem are detailed. The theory of the multiverse, popular among many physics researchers, is exposed as being a folly. 3 Infinity Without Size 4 Infinity Without Size 1.0 Countability of Sets The term "countable set" is used by the mathematics researcher Georg Cantor to describe an infinite set of numbers where we can create a one-to-one correspondence between the individual members of the set and the individual members of the infinite set of natural numbers. Similarly, the term "uncountable set" is used for a set where this correspondence cannot be created. Cantor interprets the uncountability of an infinite set as an indication that the set has a greater quantity of members than the infinite set of natural numbers, and he contends that there are an infinite amount of infinities, and each infinity has a different size. The concept that an infinite set is uncountable, in the sense intended by Cantor, may be referred to as "quantitative" uncountability. We assert that no infinite set is quantitatively uncountable. However, an infinite set can be "qualitatively" uncountable, indicating that a one-to-one correspondence cannot be made only because we do not have an ordered pattern so that given a number, the pattern will dictate the next number. For this type of infinite set, the members are inserted into the set by analyzing a particular number or category of numbers and discovering that it belongs to the set. Qualitative uncountability does not indicate greater size. We reject Cantor's notion of multiple infinities, which is based on an incorrect understanding of the concept of infinity. For the rest of this text, when the term "uncountable" or "uncountability" is used, without a modifier, we are referring to the concept of quantitative uncountability. 1.1 Rational Numbers Countability We can categorize all real numbers as rational and irrational. The rational numbers are countable. In order to prove this, we do not have to prove countability for all rational numbers. It is sufficient to prove countability for the fractions in 0, 1. The reason this proof is sufficient is: It is obvious that the integers are countable because we can create a one-to-one correspondence with the natural numbers. In regard to the countability of the rational numbers between adjacent integers, the proof that we will develop for the countability of the rational numbers in 0, 1 can be used in a parallel fashion to prove the countability of the rational numbers in all the other intervals. So we conclude that all rational numbers are countable because the union of a countable amount of sets, where each set contains a countable amount of numbers, is countable. We can show that the rational numbers in 0, 1 are countable by listing all fractions whose denominator is 2, then all fractions whose denominator is 3, ⋯. Fractions with the same denominator are listed in order of increasing numerator. Equivalent fractions such as , , ,⋯ 5 Infinity Without Size are listed only one time, in the form of the fraction with the smallest denominator. Our list appears as 123 456 789 ⋯ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕ ⋯ Another proof that the rational numbers are countable is that all rational numbers are roots of the linear equation 0, where and are integers. Linear equations can be ordered by assigning an index to each equation based on the sum of the degree of the equation (one) and the absolute value of its coefficients, so the rational numbers are countable. 1.2 Algebraic Numbers Countability Instead of categorizing all real numbers as rational and irrational, we can categorize all real numbers as algebraic and transcendental. The algebraic subcategory of the real numbers consists of all numbers that are roots of the polynomial equation ⋯ 0, where all coefficients are integers. (When we mention in the rest of this text, we are referring to this function). All rational numbers are roots of this equation. In addition, some irrational numbers are roots of this equation. For example, the radical irrational numbers that have a rational radicand and index are roots of this equation. (Numbers with a rational radicand and index are irrational, except in the case where the radicand is equal to an integer value raised to the power of the index. For example, √81 is irrational, because there is no integer that can be raised to the power of 5 and result in the value 81. In order for the expression 81 to be true, must be irrational. √81 is rational because 3 81). In addition to simple radicals such as √3, the set of algebraic radical irrational expressions also includes composite radicals such as √ √ √√ √ √ √7 5, , , and , or any expression which has a finite amount of √ √ terms involving radicals and nonradicals. We have already proven that the rational numbers subset of the algebraic numbers is countable. It is also true that the whole set of algebraic numbers, including the subset of the algebraic numbers which is irrational, is countable. This subset includes irrational radicals, and irrational numbers which are roots of where is of the fifth degree or higher and the roots cannot be expressed as radicals. The algebraic numbers can be shown to be countable, because they are roots of . Equations of this type can be organized by assigning an index to each equation based on the sum of the 6 Infinity Without Size degree of the equation and the absolute value of the equation’s coefficients, so the algebraic numbers are countable. The transcendental numbers subcategory of the real numbers consists of all numbers that are not roots of . Another way to say this is that non-algebraic numbers are transcendental. The term transcendental is derived from the fact that these numbers can be considered as transcending the equation , since they cannot be roots. All transcendental numbers are irrational. Since these numbers are not roots of , we will have to prove that this category exists. 2.0 Cantor’s Diagonal and Interval Proofs and their Refutation Cantor presented proofs showing that the set of real numbers is uncountable. Since the algebraic numbers subset of the real numbers is known to be countable, he inferred that there is an uncountable subset of the real numbers that is non-algebraic (transcendental) . We show Cantor’s diagonal and interval proofs, and then we refute the proofs. The interval proof was published in 1873, and the diagonal proof was published in 1891. Since the later, diagonal proof is more widely known, we present that proof first. 2.1 Cantor’s Diagonal Proof The proof we present is an adaptation of Cantor’s diagonal proof by contradiction.
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