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 𝑎𝑥 𝑏 0, where 𝑎 and 𝑏 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 𝑓𝑥 𝑎𝑥 𝑎𝑥 𝑎𝑥 ⋯𝑎𝑥 𝑎𝑥𝑎 0, where all coefficients are integers. (When we mention 𝑓𝑥 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 𝑥 81 to be true, 𝑥 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 𝑓𝑥 where 𝑓𝑥 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 𝑓𝑥. 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 𝑓𝑥. 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 𝑓𝑥, since they cannot be roots. All transcendental numbers are irrational. Since these numbers are not roots of 𝑓𝑥, 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. One of the differences between his proof and this adaptation is that his proof is based on the binary digits 0 and 1, and our adaptation is based on decimal numbers.

We imagine having the infinite amount of all real numbers in 0, 1 in a list, in the form of the

decimal expansion 0. 𝑎𝑎𝑎 ⋯, where 𝑎 is any digit from 0 through 9. Any number whose decimal expansion terminates is viewed as having an infinite trail of 0’s, and is considered to be equivalent to the number which shares the leading digits prior to the last nonzero digit, has a 1 subtracted from the last nonzero digit, and then has an infinite trail of 9’s. The same number can therefore be listed in two ways. We choose to list the number in the trailing 9’s version. For example, 0.4938 is viewed as the number 0.4938000 ⋯, and is written as 0.4937999 ⋯.

We make the assumption, which Cantor’s proof alleges to demonstrate is invalid, that the set of real numbers is countable. Then we can list them in a one-to-one correspondence with the natural numbers. The list will appear as

7 Infinity Without Size 1↔0.𝑎𝑎𝑎 ⋯

2↔0.𝑎𝑎𝑎 ⋯

3↔0.𝑎𝑎𝑎 ⋯

⋮

We now create a number 0. 𝑏𝑏𝑏 ⋯, where 𝑏 is any digit other than 𝑎 and other than 0, 𝑏 is any digit other than 𝑎 and other than 0, 𝑏 is any digit other than 𝑎 and other than 0, ⋯. (Since this number is built based on the fact that each 𝑏 digit is different from the 𝑎 digit, and the 𝑎 digits form a diagonal string through the list of numbers, this strategy can be called the diagonal argument). When we examine 0. 𝑏𝑏𝑏 ⋯ we observe that although it is in 0, 1, it is different than every number on the list, because its digit in the 𝑏 position differs from the digit in the 𝑎 position. We have discovered a number external to the list. Therefore, our assumption that all numbers in 0, 1 can be written in a list consisting of these numbers in a one-to-one correspondence with the natural numbers is not true, so the set of real numbers in 0, 1 is uncountable, in the sense that the set of real numbers is more numerous than the set of natural numbers

Since the real numbers are uncountable, and the algebraic numbers subset, which includes rational numbers and a subset of irrational numbers, is countable, we further conclude that a category called transcendental numbers exists, and is uncountable.

Cantor includes all real numbers in the list that he uses for his diagonal argument, instead of just including transcendental numbers in the list and showing that there is a transcendental number that is not in the list, proving that the transcendental numbers are uncountable. The reason for including all real numbers in the list is because he uses his proof not just to prove that the transcendental numbers are uncountable, but also to prove that they exist. Therefore, at the beginning of the proof, we cannot list only the transcendental numbers, because at that time we are not working with the knowledge that they exist.

Also, even if we could assert at the beginning of the proof that the category of transcendental numbers exists (which we could know, since we are aware of numbers which are not algebraic, such as Liouville numbers, described in Section 3.1), we can understand why Cantor’s proof needs to include the algebraic numbers along with the transcendental numbers. If we only build a list of transcendental numbers, then when we develop a diagonal-based number that is not on the list, we can allege that this does not prove that our list of transcendental numbers is incomplete, because the created number may be an algebraic rational number, with an infinite string of repetitive pattern digits, or an algebraic irrational number, with an infinite string of nonrepetitive pattern digits. Therefore, we have to include the algebraic numbers, so we build a list of real numbers. Now, since we have included all real numbers in the list, the presence of a new number, which does not match any real number on the list, proves that the real numbers are

8 Infinity Without Size uncountable, and since the algebraic numbers are countable, the transcendental numbers must be uncountable.

When building a list of real numbers, we can understand why Cantor stipulates that the diagonal- based number should not include any 0’s. If 0’s were permitted, we could suggest that the new number has an infinite trail of 0’s, and is identical to a similar number with an infinite trail of 9’s, so we have not demonstrated the existence of a new number (since Cantor maintains that 0.4938000 ⋯, for example, is equal to 0.4937999 ⋯).

2.2 Refutation of Cantor’s Diagonal Proof

Cantor’s diagonal proof is not valid, as we will show.

When we analyze the justification for considering a string of numbers following a decimal point as being a number, we conclude that the digits obtained by the diagonal argument do not qualify as a number. Therefore, we have not discovered a number that is not on the list, and we have not proven that the real numbers are uncountable.

A string of digits following a decimal point is a number or a precursor to a number according to the presence of the conditions that we now list. (The term precursor will be explained).

1. If there is a terminating string of digits, the expression is a rational number.

2. If there is an infinite string of digits following the decimal point with a repetitive pattern, the expression is the precursor to a rational number limit. As an example, 0.142857142857 ⋯ is the precursor to the rational number , which cannot be written precisely as a decimal expansion. The precursor is not a number, but is the partial sum of an infinite series which we consider, in an imaginary manner, to be the number closest to the limit. (See The Infinity Numbers, by this author). We can create any repetitive pattern number by arbitrarily composing a repetitive pattern of digits. The amount of digits that repeat can be one or any number greater than one. We can determine the limit of the resulting infinite string of digits that we have created using the method demonstrated in The Infinity Numbers in Section 6.1.

3. If there is a known infinite string of digits following the decimal point with no repetitive pattern, this expression is the precursor to a known irrational number limit. As an example, the partial sum 3.141592654 ⋯ is the precursor to the irrational number 𝜋, which cannot be written precisely as a decimal expansion. We choose how many digits to write as a representation of the precursor based on the level of precision we wish to achieve. We cannot arbitrarily select digits with a nonrepetitive pattern and consider the string to be a precursor to an irrational number. There must be basis for the number. There may be a geometric basis for the number, such as in the case of π, or there may be a function basis

9 Infinity Without Size for the number, such as in the case of the irrational numbers generated by the radical function and the log function. We obtain the digits of the precursor by means of an infinite series.

4. If there is an infinite string of digits following the decimal point with no repetitive pattern, and the string has been created by use of an algorithmic constructor, we know that the number is irrational, but we are unable to identify the limit of this string in order to be able to use it in calculations. (In contrast, in the case of 𝜋 and other known irrational limits, we have a known limit that we can use in calculations). As an example, the string of digits 0.010010001 ⋯ is built using an algorithm that specifies that we start with a section containing 01, then concatenate successive sections without bound, where each section includes an additional 0 before the 1. Although we cannot identify the limit of this string of digits, we know that the expression is a precursor to a valid number, because there are an infinite amount of points on the number line, so one of the points has to be represented by this string of digits.

However, Cantor’s diagonal-based expression does not meet any of these criteria.

The first criterion is not met because the digits of the expression cannot be written in their entirety. The second criterion is not met because the construction of the string does not embody a repetitive pattern. (We also note that even if the expression would meet the first or second criterion, Cantor’s proof would not be salvaged. On the contrary, the appearance of this number would contradict Cantor’s view, because although Cantor claims that an irrational number can be found which is not on the list, he agrees that no new rational number can be discovered beyond the numbers on the list). The third criterion is not met because there is no geometric or function basis for the expression.

One may claim that the fourth criterion is met, because there is an algorithmic constructor that consists of making a change to each digit of the current diagonal for the purpose of constructing a new number. It may seem that Cantor’s proof by contradiction is valid. However, we can show that the proof is not valid.

These are the circumstances where a proof by contradiction is valid.

 The original statement is arbitrary (only an assumption).  The succeeding steps are unquestionably true.  The conclusion of the succeeding steps convincingly contradicts the original statement.

In the case of Cantor’s diagonal argument the original statement is that the numbers in 0, 1 can be written in a list consisting of these numbers in a one-to-one correspondence with the natural numbers. Although this is not entirely true, because the set of real numbers is qualitatively uncountable (we cannot find all the numbers), it is true that there is no number that cannot be included in the list of real numbers (the set of real numbers is not quantitatively uncountable).

10 Infinity Without Size (The distinction between the two types of uncountability is discussed in Section 1.0). Since there is no number that cannot be included in the list, we accept the original statement as being a fact, and not an assumption. We base our confidence on the correct understanding of infinity, which rejects the notion that there are sized infinities.

The succeeding steps lead to the conclusion that there is a string of digits that is not included in the list of real numbers. These steps are obviously true. However, since this string is only

defined by describing what each digit cannot be (the digit in the 𝑏 position is different from the digit in the 𝑎 position), and not by describing what it is, we deem these digits to be merely an infinite collection of digits that are not a precursor to any limit. This collection of digits is not a number, and is not an indication of an associated limit. Therefore, we have not found a contradiction between the conclusion and the original statement, so the original statement, which is a fact and not an assumption, remains unchallenged.

(We note that there is no reason for Cantor to include the endpoints of the interval 0, 1 in his attempted proof. 0 and 1 are members of the set of integers, which is known to be countable. The attempted proof should be restricted to the open interval 0, 1).

2.3 Reinterpreting the List Used for Diagonal Argument

According to Cantor’s diagonal proof, an expression with trailing 9’s is equivalent to a similar expression with trailing 0’s, and all rational numbers can be written as a decimal expansion. Both of these ideas are mistakes. They are both caused by the fundamental error of equating a precursor with its limit, symbolized by the erroneous expression 0.999 ⋯ 1.000 ⋯, or 0.999 ⋯ 1. The correct view is that 0.999 ⋯ 1, and that some numbers cannot be written as a decimal expansion (see The Infinity Numbers). For example, 0.4937999 ⋯ is the precursor to 0.4938000 ⋯, but the two expressions are not equal. Also, cannot be written as a decimal expansion. We can represent approximately by 0.333 ⋯, but 0.333 ⋯ is only the precursor to . 0.333 ⋯ is not equal to .

However, this fundamental error made by Cantor is not the basis for refuting his proof. We could leave his list unchanged, and just modify our understanding of the list. We could then assert the same faulty diagonal proof.

Our conceptual modification to the list would consist of recognizing that all the expressions that are on the list are precursors, representing both rational and irrational numbers.

 For rational numbers which can be written as a string of terminating digits, instead of listing the numbers as strings with an infinite trail of 0’s, we list the numbers as represented by their precursors, which consist of similar strings with an infinite trail of 9’s. For example, 0.25000 ⋯ is written as 0.24999 ⋯. We keep the requirement that

11 Infinity Without Size the diagonal-based digits not be 0. If we would allow 0’s, then we must consider that the created diagonal string of digits may have infinite trailing 0’s. Then that string would be the limit of a precursor on the list, and we have not discovered a new number. In order to ensure that a new number is created, we stipulate that 0 should not be chosen for any digit in the digit string.  Rational numbers that cannot be written as a decimal expansion are represented by their precursor. For example, is written as 0.333 ⋯.  Irrational numbers are represented by their precursor, which consists of a string of nonrepetitive pattern digits. For example, 𝜋3 is written as 0.141592654 ⋯.

Based on this modified understanding of the list, we could still try to assert that there is a proof that the real numbers are uncountable. However, we have shown in Section 2.2 that the proof is invalid.

2.4 Cantor’s Interval Proof

We describe Cantor’s interval proof for the uncountability of real numbers. This expanded and narrative description is an adaptation of his proof.

We can create an infinite ordered list of real algebraic numbers. Since the list is ordered, it can be placed in a one-to–one correspondence with the set of natural numbers. However, we will discover that there are an infinite amount of numbers that are not present in this list, which means that they are not algebraic, so we will call them transcendental (beyond algebraic). We conclude that the set of real numbers has two subsets: the set of algebraic numbers and the set of transcendental numbers.

The set of transcendental numbers cannot be ordered by the same scheme as we used in the case of the real algebraic numbers, or by any other scheme. Since ordering is not possible for this subset, it is not possible to place the elements of this set in a one-to-one correspondence with the set of natural numbers.

Cantor uses the fact that the set of transcendental numbers exists, and cannot be placed in a one-to-one correspondence with the set of natural numbers, as a basis for his assertion this set is uncountable, in the sense that its members are more numerous than the members of the set of natural numbers. He therefore concludes that the set of real numbers, which includes the transcendental numbers, is uncountable by extension.

Before presenting Cantors’ proof that there are infinite transcendental numbers, it would be useful to show that we can find an infinite amount of numbers that are not present in the set of rational numbers. These numbers are irrational.

12 Infinity Without Size We create an infinite list of rational numbers in the interval 0, 1 that are ordered by increasing magnitude of the numerator within increasing magnitude of the denominator. We omit equivalent fractions. The list appears as

1 1 2 1 3 1 2 3 4 ⋯ 2 3 3 4 4 5 5 5 5 We will find an infinite amount of numbers in the interval 0, 1 that are not present in our list.

We move through the list and select the first pair of numbers that are in the interior of the interval 0, 1. We now have a new, narrower interval. Then we move further through the list, and we select the first pair of numbers that are in the interior of the current, narrower interval. This pair delineates a new interval. We repeat the process of interval creation infinite times, forming an infinite group of nested intervals.

We obtain these intervals (the equivalent decimal values are approximate):

, → 0.33, 0.50

, → 0.40, 0.42

, → 0.411, 0.416 ⋮ ⋮

In our construction of the list, the amount of intervals is infinite, because there will always be two numbers present at some subsequent position in the list, interior to the previous interval, to use as the endpoints of a new interval.

The numbers comprising the left endpoints form an infinite increasing sequence of numbers

𝑎, and the group of numbers comprising the right endpoints form an infinite decreasing sequence of numbers 𝑏. For any position 𝑘 in the list, the distance between 𝑎 and 𝑏 is smaller than the distance between 𝑎 and 𝑏.

We now show that lim 𝑎 lim𝑏. → →

We will call lim 𝑎 𝐿 and lim 𝑏 𝐿. For any value 𝜖, regardless of how small 𝜖 is, we → → can find an index 𝑘 such that for the interval 𝑎,𝑏, it is true that |𝑏 𝑎| 𝜖, so lim 𝑎 𝐿 and lim 𝑏 𝐿, so 𝐿 𝐿 𝐿. → →

13 Infinity Without Size We have shown that 𝑎 and 𝑏 have a common limit. This number is within the interior of all the intervals. This limit cannot be contained in the list of rational numbers that we have created, because no number in the list can be in all the intervals. This is true because if a number is in the list, then if it is chosen as an endpoint of an interval, that places the number outside that interval and the following intervals. If it is bypassed as an endpoint, that means that it is ineligible to be an endpoint because it not in the interior of the previous interval, so the number is at least not in the previous interval and the succeeding intervals, and possibly also not in some or all prior intervals within 0, 1. Therefore, the limit, which is in the interior of all the intervals, cannot be in our list of rational numbers.

We have discovered a number 𝐿 that is in the interval 0, 1, but is not contained in our list of rational numbers. This number is irrational. (This number has been shown to be the value √2 1).

Since we can apply the same reasoning to an infinite sequence of intervals within 0, 1, and thereby produce infinite 𝐿’s that are irrational, we have discovered an infinite amount of numbers that are irrational.

Suppose we develop a list of rational numbers where the amount of intervals is finite, because the construction of this hypothetical list ensures that after a particular interval is specified, the remaining numbers in the list, with the exception of at most one number, are outside of the last interval. In this case we do not have two new interior numbers that are needed to create a new interval. Then the infinite amount of numbers within the final interval are not present in the list of rational numbers, so we have discovered an infinite amount of numbers that must be irrational.

We can similarly apply our analysis to an infinite list of algebraic numbers within the interval 0, 1, ordered based on their algebraic identity (as discussed in Section 1.2). If the construction of the list allows us to form an infinite amount of intervals, we will find a number 𝐿, which is the limit of the sequence 𝑎, which cannot be in the list of algebraic numbers. As in the case of the rational numbers, since we can apply the same reasoning to an infinite sequence of intervals within 0, 1, and thereby produce infinite 𝐿’s that are not algebraic values, we have discovered an infinite amount of numbers that are not algebraic.

If the construction of the list dictates that there is a finite amount of intervals, then we have a final interval, whose interior numbers are not in the list. Then we conclude that we have discovered an infinite amount of numbers that are not algebraic.

The numbers that are not algebraic are by definition transcendental.

It is not possible to list all the real numbers such that each number has a one-to-one correspondence with the set of natural numbers, because of the presence of the subset of transcendental numbers, where no ordering scheme is possible. According to Cantor, we can

14 Infinity Without Size infer from the inability to create a one-to-one correspondence between the set of real numbers and the set of natural numbers that the set of real numbers is uncountable, in the sense that it is more numerous than the set of natural numbers.

2.5 Refutation of Cantor’s Interval Proof

We cannot create an infinite ordered list that will include all the real numbers and have a one-to- one correspondence with the set of natural numbers, because the transcendental numbers subset cannot be ordered. But there is no basis for inferring from this inability that the set of real numbers is uncountable, where the term uncountability is intended to indicate that a set is more numerous than the set of natural numbers.

However, the interval proof achieves a measure of success by convincingly demonstrating the existence of transcendental numbers.

2.6 Comparing the Diagonal and Interval Proofs

The diagonal proof’s strength is that it assures, by virtue of the construction of the diagonal, that there is a string of digits that is external to any list of real numbers, seeming to prove that the real numbers cannot be counted.

However, the arbitrariness of the string’s construction deprives it of being called a number, and we consider the string to be an amalgamation of digits that has no numerical value.

The interval proof’s strength is that an infinite amount of actual numbers besides the algebraic numbers have been discovered.

However, there is no basis for inferring the uncountability of the set of real numbers, where the term uncountability is intended to indicate that a set is more numerous than the set of natural numbers.

3.1 Irrational Numbers are Not Uncountable

The set of transcendental numbers is a subset of the set of irrational numbers, which is a subset of the set of real numbers. It is implicit in the fact that the presence of transcendental numbers does not make the set of real numbers uncountable, that its presence does not make the set of irrational numbers uncountable.

Numbers are determined to be transcendental by examining a number, such as 𝜋, or a category, such as the log function (which produces transcendental numbers in most cases where the input is rational), and discovering numbers that belong in the set of transcendental numbers. However, we do not have an algorithmic method of formulating an infinite set of transcendental numbers,

15 Infinity Without Size to be organized as an ordered infinite set of numbers, as we have for the set of rational and algebraic numbers. Therefore, we cannot align the transcendental numbers with the list of natural numbers. But the inability to count the set of transcendental numbers is not due to the set being more numerous in comparison with the natural numbers. Rather, we cannot count what we have no cognition of. We describe the set of transcendental numbers as qualitatively uncountable. Since we have the ability to count the transcendental numbers that we are aware of, the set of transcendental numbers is not quantitatively uncountable.

We will show how countability is possible for types of irrational numbers, including irrational numbers that are transcendental. From this discussion we will see that although we cannot refer to the set of irrational numbers as countable, there is no trait in the irrational numbers which makes them quantitatively uncountable. As long as we can find subsets of irrational numbers that we can cognitively access, those numbers are countable.

We now show subsets of the irrational numbers that are countable. These subsets are one of four types:

1. The set of irrational numbers that can be generated from rational numbers using various functions. Since the rational input numbers are countable, their corresponding irrational output numbers are also countable. The functions that produce this rational to irrational metamorphosis include:

a. The radical function, where the radicand and index are rational, both in the case of a square root where the radicand is not a perfect square, and in the case of any other 𝑛th root function where the result is irrational. The irrational outputs are countable based on their rational inputs (and also based on the fact that they are roots of 𝑓𝑥, as described in item 2).

b. The trigonometric functions, where input values are rational multiples of π, and the output values are irrational. Although the inputs are irrational because of the presence of 𝜋, the irrational outputs can be directly connected to the rational multipliers of 𝜋. For the first quadrant, the only output values that are not irrational are, depending on the particular trigonometric function, values whose inputs are some combination taken from 0, , , . An example of this exception is sin , whose value is the rational number . Although we are treating irrational trigonometric numbers as a separate group, we note that since they are all radicals, they are also included in group 𝑎. The infinite amount of irrational numbers produced by the trigonometric functions are countable by noting the correspondence between each irrational number and its rational antecedent.

c. The natural logarithm function, where the input values are rational and the output values are irrational. (Of the three groups 𝑎, 𝑏, 𝑐, this is the only group whose numbers are also

16 Infinity Without Size transcendental). These irrational numbers are countable based on their derivation from countable rational numbers.

2. The set of irrational radicals, which are roots of 𝑓𝑥. As discussed in Section 1.2, equations of the type 𝑓𝑥 can be organized according to their index, so their roots are countable.

3. The set of irrational numbers, such as 𝜋 and 𝑒, which are constants based on geometric and other computational considerations. Some of these numbers have been proven to be transcendental.

4. The set of irrational numbers that can be developed using an algorithm to create their precursors, such as the number whose precursor is 0.010010001 ⋯, where an additional 0 is placed before each succeeding 1, and the Liouville numbers, such as the number whose precursor is 0.1100010 ⋯, where the 1's occur in the decimal positions numbered 1!, 2!, 3!, ⋯. The Liouville numbers have been proven to be transcendental.

We do not have an awareness of an infinite amount of functions, geometric considerations, and algorithms that can produce non-algebraic irrational numbers. It is only for this reason that we say that the set of irrational numbers cannot be counted. The term "uncountable" is erroneously used in regard to infinite sets to indicate greater quantity in comparison with the infinite set of natural numbers. This interpretation of uncountability is meaningless, because infinite sets do not have any size. We assert that the set of irrational numbers, which includes the transcendental numbers, and therefore the set of all real numbers, is not quantitatively uncountable, but only qualitatively uncountable.

4.1 Cantor’s Power Set Proof

Cantor attempted to prove that the power set of the infinite set of natural numbers has a greater cardinality than the infinite set of natural numbers by this proof.

We begin by examining the case of the power set of the finite set of numbers 1, 2, 3 . Its power set is ∅, 1, 2, 3, 1, 2, 1, 3, 2, 3, 1, 2, 3. We reverse the order of 2 and 3 and create this list:

1 2 3 4 5 6 7 8

↕ ↕ ↕ ↕ ↕ ↕ ↕ ↕

∅, 1, 3, 2, 1, 2, 1, 3, 2, 3, 1, 2, 3

We will call each power set element a record, and the natural number that corresponds to a particular power set record a record number. We call record number 3 a matching record

17 Infinity Without Size number because the record contains its record number. All other record numbers are nonmatching numbers, because their records do not contain their record number.

In order for the power set of the infinite set of natural numbers to have the same cardinality as the infinite set of natural numbers, there must be a one-to-one correspondence. We will show that this correspondence does not exist, and therefore the cardinality of the two sets is not the same.

There is an infinite amount of record numbers that do not match any number in their record. Since the power set contains records representing all possible combinations of the natural numbers, one of the records in the list contains only all the nonmatching record numbers across the entire power set. The amount of numbers in this record is infinite. We now consider the question of what is the record number of the record containing the nonmatching numbers. Let us say that it is 50. Then 50 cannot be a number in this record, because if it were, then the number of this record, 50, would not be a nonmatching record number. So we decide that 50 is not in the record. But if 50 is not in the record, then record 50 is a nonmatching record number, so this number should be in the record. There is a contradiction. So we conclude that the amount of records in the power set exceeds the amount of numbers in the set of natural numbers, and there is no one-to-one correspondence. Then we can assert that the record containing only all the nonmatching record numbers is beyond the natural numbers, so the cause for the contradiction disappears.

4.2 Refutation of Cantor’s Power Set Proof

A power set of a finite set is a collection of elements representing all numerical combinations of all members of the finite set. Each numerical combination appears in the power set as one element. Each element has the status of a set. However, in the case of a power set of an infinite set, we cannot refer to "all numerical combinations of all members of the infinite set" in a literal sense. "All" indicates a total aggregate of items, and is a finite concept. The meaning of infinity is an increase without bound, which implies an unlimited potential for the presence of eligible items. In an infinite set, the items exist only in an inductive sense. The presence of item 𝑎 implies the presence of item 𝑎.

In order to define the power set of an infinite set, we need a different interpretation of "all numerical combinations" and "all members". A suitable interpretation will emerge from our discussion.

We will use the infinite set of natural numbers as our case of an infinite set.

In the case of a power set of a finite set of natural numbers, we can use an algorithm to produce the power set. For example, for the finite set of natural numbers 1, 2, 3, we can create a power set by inserting the empty set ∅ as the first member of the power set. Then we select the

18 Infinity Without Size individual natural numbers and place each of them as a set within the power set, so we have ∅, 1, 2, 3. Then we take the combination of the first two natural numbers of the original set and append this combination as a new member of the power set, so we have ∅, 1, 2, 3, 1, 2. If we continue in this manner, we emerge with this power set:

∅, 1, 2, 3, 1, 2, 1, 3, 2, 3, 1, 2, 3

If we subsequently rearrange any of the members of the power set, the resulting power set would be the same as the original power set, but the order of the members would be different.

We can also arbitrarily form all numerical combinations of the natural numbers that are found in the original set, and collect these combinations into a power set. If we do that, we will obtain the same power set as the one that is created methodically, although the two images of the power set will likely differ in the ordering of their members.

In order to discuss the power set of the infinite set of natural numbers, we first need to present the interpretation of "the set of all natural numbers":

1. The set contains only natural numbers.

2. There is no natural number which is excluded from the set.

We cannot use an algorithm to create a power set of the set of natural numbers. We would expect the combination 1, 2 to appear in the power set after the last individual number of the set of natural numbers, but there is no "last" individual number in the infinite set of natural numbers.

In order to be able to say that the power set of the set of all natural numbers contains all numerical combinations of all natural numbers, we employ this interpretation:

1. The power set contains only numerical combinations of only natural numbers.

2. There is no numerical combination or natural number which is excluded from the power set.

We cannot say that all numerical combinations of all natural numbers are actively included in the power set, because an infinite amount of items cannot be collected and placed in a group. However, any numerical combination and any natural number are eligible for entry into the power set, and none can be excluded.

In addition to describing the power set as consisting of all numerical combinations of the set of all natural numbers, we can also define a particular element of the power set using a descriptive characterization of the element.

19 Infinity Without Size For example, we can describe one of the elements in the power set as being a collection of all the even numbers. This collection is an infinite set which contains only even numbers, and no even number is excluded.

We can describe an element in the power set as being the set 1, 5, 9, ⋯ . This is an infinite set that:

1. contains only odd numbers beginning with 1 and continuing with numbers four units ahead of the previous number, and

2. does not have any exclusion of the type of number described in item 1.

We now examine the part of Cantor’s power set proof that claims that there is a member of the power set which contains a list of all the nonmatching record numbers across the power set. According to our interpretation of "all" when the word is used in regard to infinity, this descriptive characterization means a set whose elements all share the characteristic that they are nonmatching record numbers, and there is no nonmatching record number excluded.

However, contrary to the implication of the interpretation of "all", either there is an element that does not share the characteristic, or there is an exclusion, and therefore this descriptive characterization is inherently invalid. We now explain.

Suppose we identify record number 50 as being the set whose elements consist of all the nonmatching record numbers. Then if the set contains the number 50, we would be incorrect in describing this set as containing only elements that are nonmatching record numbers, since record number 50 contains the number 50. But if the set does not include the number 50, then record number 50 is a nonmatching record number, and should have been included. Since it was not included, it is an exclusion. So neither record number 50, nor any other record, can be accorded the descriptive characterization of containing all the nonmatching record numbers.

However, since all numerical combinations must be present in the power set of the set of natural numbers, it is possible for record number 50 to have either of these appearances:

1. Contains all nonmatching record numbers, and also number 50:

Then we recognize that the numbers contained in record number 50 are all nonmatching record numbers, except for the number 50, which is contained in record number 50 and is therefore not a nonmatching record number.

2. Contains all nonmatching record numbers, except for the number 50:

Then we acknowledge record number 50 as having all the nonmatching record numbers except for its own number.

20 Infinity Without Size Although an infinite set of descriptive characterizations is possible when describing members of the power set, we cannot say that this infinite set contains all descriptive characterizations, because that expression would mean that:

1. the set consists only of descriptive characterizations of combinations of the natural numbers, and

2. there is no descriptive characterization which is excluded from the set.

The second condition cannot be met, because there are exceptions, such as the descriptive characterization of one of the members of the power set as containing all nonmatching record numbers.

It is not necessary for the set of descriptive characterizations defining elements contained in the power set to be comprised of "all descriptive characterizations". Although the power set has members with an infinite amount of descriptive characterizations, this does not mean that every variation of descriptive characterization is valid and must be present. Infinity implies an increase without bound, and not the presence of all variations. Therefore, if a descriptive characterization that we conceive of is self-contradictory, then that descriptive characterization is rejected as a member of the set of descriptive characterizations. This rejection does not indicate any loss of inclusiveness in the list of numerical combinations which comprises the power set, because it remains true that there are no numerical combination exclusions.

The description of a set as containing all nonmatching record numbers is a self-referential description. Self-referential statements are apt to produce contradictions. An example of an invalid self-referential statement also occurs in the field of logic when we attempt to determine the truth value of the statement "I am a liar". If the statement is true, then the person is lying, so the statement is false. Then the person is not a liar, and only tells the truth. But if the person only tells the truth, then the statement is true, and the person is a liar. So if the statement is true, it is false, and if it is false, it is true. This self-referential statement is invalid, just as the self- referential descriptive characterization regarding all nonmatching record numbers is invalid.

The contradiction conjured by Cantor disappears, because there is not, and does not have to be, a set which contains all nonmatching record numbers. We assert that Cantor's concept of multiple infinities, which uses the power set proof as a basis, is invalid.

5.0 A Single Infinity Without Size

We now summarize our conclusions regarding countability and infinity.

Irrational numbers that emerge from functions are countable because the origin of each irrational number is a countable rational number. The irrational numbers that are solutions to 𝑓𝑥, and irrational numbers that are geometric constants or that are constructed from an algorithm, are

21 Infinity Without Size also countable. Our inability to count other irrational numbers is not due to uncountability of these numbers, but is due to the fact that we do not have any cognizance of these numbers. We cannot count numbers whose existence we are unaware of. Therefore, we assert that the set of irrational numbers is not quantitatively uncountable, but rather qualitatively uncountable. The cardinality of the irrational numbers is the same as the cardinality of the natural numbers. We also assert that the cardinality of a power set is equal to the cardinality of the original set.

Our understanding of infinity is "continuing without bound". An infinite set is characterized by its potential for the unbounded appending of members. An infinite set is not a completed group of all possible members. Therefore, all infinite sets have the same cardinality, which is the cardinality of no size. The multiplicity of cardinalities alleged by Cantor does not exist.

6.0 Identification of Transcendental Numbers

We may wonder why it is necessary for mathematicians to develop proofs showing that particular numbers are transcendental. We may argue that no irrational number can be a root of 𝑓𝑥 𝑎𝑥 𝑎𝑥 𝑎𝑥 ⋯𝑎𝑥 𝑎𝑥𝑎 0, since the right side of the equation is rational.

However, an irrational number can conceivably be a root of this equation, as we explain now.

Irrational radical numbers can be roots either because an exponent can undo a radical, such as in the case 𝑥 20, where 𝑥√2, or because the addition or subtraction of two irrational terms in 𝑓𝑥 could cause the combination to become rational.

An example demonstrating how subtraction can cause an expression involving irrational numbers to be rational is the case of the equation 8𝑥 6𝑥10. The root of the equation is 𝑥 sin sin 10°. sin , like almost all trigonometric values, is an irrational radical, and yet the right side of the equation is rational. Although 6𝑥 and 8𝑥 are both irrational, the subtraction of 6𝑥 from 8𝑥 produces a rational result.

Some irrational numbers, such as 𝜋 and 𝑒, can be proven to be transcendental, so they cannot be roots. Therefore, we conclude that they cannot be radicals.

7.1 Niven’s Precepts on Irrational Numbers and Measure Zero

We present the mathematical analysis of Ivan Niven in the beginning of his book Irrationals (Mathematical Association of America, 1956) p. 2-5.

We label a subset of the real numbers as having measure zero if all the points on the number line which represent the numbers of the subset can be viewed as having an interval adjacent to both

22 Infinity Without Size sides of the point in a manner such that the total size of all the intervals is of arbitrarily small length.

1. The set of rational numbers has measure zero, as we will now show. We surround each rational number ,1ℎ𝑘, with an interval . 2lim ∑∑ 2lim∑ , → → which by the zeta function ζ2 gives us 2 , which is less than 4. Therefore, 2lim∑∑ 4𝜖. This number can be made arbitrarily small by choosing a small → enough value for 𝜖. We have proven that the positive rational numbers in 0, 1 have measure zero. It can be shown that this proof can be extended to all rational numbers greater than 1, and then to the negative rational numbers and zero.

2. The set of rational numbers is countable, because there is an ordering scheme that includes all members of the set.

3. Any countable set of real numbers has measure zero, which can be shown by surrounding each number with the interval , where 𝑘 refers to the number’s position in the ordering scheme. The limit of the geometric series 2lim ∑ is 2𝜖. 2𝜖 can be made arbitrarily small. → Since the set of rational numbers is countable, we can use this method to prove that the set of rational numbers has measure zero, in addition to the method using the interval , shown previously.

4. The set of real numbers cannot be countable, because then it would have to have measure zero. Niven says it is intuitively clear that the real numbers do not have measure zero, and he claims to have a rigorous proof which he says he chooses not to provide in this discussion. Since the set of rational numbers is countable and has measure zero, the set of irrational numbers cannot also be countable and have measure zero, because then the set of real numbers, which consists of the set of rational numbers and the set of irrational numbers, would have to be countable and have measure zero. So the set of irrational numbers has been proven to be not countable and to not have measure zero. Then it follows that the set of real numbers is not countable.

Niven shows that the distance between √ and any rational number is , so √ cannot be within an interval of surrounding the rational number when we set 𝜖 to be or smaller.

23 Infinity Without Size 7.2 Dismissal of Niven’s Ideas

We now dismiss Niven’s ideas regarding irrational numbers and measure zero.

We contend that the actual intuitive truth regarding the characteristic of measure zero in relation to real numbers is the opposite of the supposed intuitive truth as asserted by Niven. It is intuitively true that the real numbers can be surrounded by arbitrarily small intervals, whose sum converges to zero. So the set of real numbers has measure zero, and therefore it is clear that the set’s components, which are the set of rational numbers and the set of irrational numbers, have measure zero. A proof that the real numbers have measure zero will be shown later.

Although Niven claims that he that he has a rigorous proof that the real numbers cannot be surrounded by intervals of arbitrarily small total length, he did not present his proof or refer to another location where his proof is presented, so we disregard his claim.

Niven uses the supposed fact of the uncountability of real numbers to prove the uncountability of the irrational numbers, and then he follows with using the uncountability of the irrational numbers to prove the uncountability of the real numbers. This reasoning is circular logic.

Every irrational number exists within any interval of a rational number, no matter how small the interval is. As an example, the irrational exponential constant 𝑒 is the limit of the infinite series of rational terms ∑ ⋯. The partial sums, which are all rational, ! ! ! ! converge to 𝑒. Since all the partial sums are rational, and the limit is irrational, we see that a rational partial sum can be as close as we wish to the limit 𝑒, and the interval separating the rational partial sums and the irrational limit approaches zero.

Niven showed that the distance between √ and any rational number is greater than . However, lim 0, so √ can be as close to a rational number as we wish, as long as 𝑘 is → large enough. In The Infinity Numbers, Table 7 in the appendix shows an infinite series of rational terms which converges to √2.

Even though the set of real numbers is qualitatively uncountable, because we do not have an ordering scheme that we can impose on it, we can show that it has measure zero. In The Infinity Numbers, we prove the nonexpandability principle, which states that 𝑛𝜖 𝜖, 1𝑛∞. Using this principle, we can surround the real numbers with intervals of 𝜖, and the intervals will add to 𝜖. Since 𝜖 can be chosen to be as small as desired, the sum of the intervals approaches zero. We conclude that the real numbers have measure zero.

Niven contends that since the rational numbers have measure zero and the irrational numbers do not have measure zero (according to his view), almost all real numbers are irrational. However, even if we were able to show that irrational numbers have intervals surrounding them whose sum

24 Infinity Without Size cannot be made arbitrarily small under any circumstances, and that this characteristic does not apply in the case of rational numbers, there would be no indication that the set of irrational numbers are more numerous than the set of rational numbers.

Every irrational number has another irrational number adjacent to it, with a closeness that is greater or smaller by a rational value 𝜖, which can be made as small as we wish. If we add an irrational 𝜖 to an irrational number, the result can be either rational or irrational, depending on whether the irrational 𝜖 is an 𝑖-complement of the original irrational number, as described in The Infinity Numbers.

Every rational number has a number adjacent to it which is greater or smaller by a value 𝜖, which can be made as small as we wish. The rationality of the resulting value depends on the rationality of 𝜖. If 𝜖 is rational, then the new number will be rational. If 𝜖 is irrational, then the new number will be irrational.

8.1 Flawed Reason for Accepting the Axiom of Choosing

The mythical axiom of choosing (axiom of choice), which is based on the imaginary choosing function (function of choice), has the unique status of being an axiom which is accepted by conventional mathematical doctrine not because of any conviction of its reasonableness, but solely due to its usefulness, and despite the declared or undeclared recognition, even by its proponents, that it is fictional.

8.2 History of the Axiom of Choosing

We review the history of the development of the axiom of choosing. In the quotations cited in this section, italics are omitted.

In 1883, Cantor introduced the idea of a well ordering theorem, which asserted that all sets, including the set of real numbers, can be well ordered. Cantor wrote, "(The fact that) it is always possible to arrange any well-defined set in the form of a well-ordered set is, it seems to me, a very basic law of thought, rich in consequences, and particularly remarkable in virtue of its general validity".

A well ordered infinite set, as subsequently understood, has the property that there is a "least" element in all of its subsets. In the context of this description, "least" does not have to mean least numerical value, but can also mean an item that is necessarily (without arbitrary selection) the first element in the list.

There is an obvious obstacle to well ordering, which can be seen if, as an example, we examine one of the subsets of the set of real numbers, 𝑥| 2 𝑥 3. In this case, a well ordering does not appear possible. This set has no numerically least element, since for any element we select,

25 Infinity Without Size there is another element smaller in value, but still greater than 2. There is also no element which qualifies by necessity (without arbitrary selection) for nonnumerical first. Cantor did not explain how this obstacle can be overcome.

In 1904, the mathematics researcher Ernst Zermelo asserted that the well ordering theorem can be proven based on an axiom he developed, which may be called the axiom of choosing. He declared that there is a function, which may be called the choosing function, which can operate on each of the infinite amount of disjoint unordered finite subsets of the infinite set of real numbers by systematically and nonarbitrarily choosing a first member, and a second member, and a third member, ⋯, thereby establishing an order within the subset. All choosings in all subsets occur simultaneously, similar to what occurs in the case of an infinite amount of sets that each contain a numerically least value, where all the least values intrinsically and simultaneously stand out from the other members of the set. If we take the union of all the well ordered subsets of the set of real numbers which have been well ordered in this manner, we find that the set of real numbers has been well ordered, because each subset has a first value, which we call the "least" value.

In 1908, Zermelo acknowledged that the axiom of choosing concept, as he originally presented it, is problematic, because no element has any distinguishing feature which sets it apart from the other elements in the subset, so there is no actual function which can automatically recognize and choose one element from among all the elements in the group. Nevertheless, he insisted that the axiom of choosing must be accepted, because for a variety of problems the axiom of choosing is the only way to achieve solutions which conform with results that are expected according to Cantor's multiple infinities theory.

Also in 1908, Zermelo compared the acceptance of the axiom of choosing to the acceptance of Euclid's parallel postulate, which is used even though it cannot be proven. He wrote, "Banishing fundamental facts or problems from science merely because they cannot be dealt with by means of certain prescribed principles would be like forbidding the further extension of the theory of parallels in geometry because the axiom upon which this theory rests has been shown to be unprovable".

Zermelo also wrote in 1908 in regard to the axiom of choosing, "⋯ even in mathematics unprovability, as is well known, is in no way equivalent to nonvalidity, since, after all, not everything can be proved, but every proof in turn presupposes unproved principles. ⋯ this axiom, even though it was never formulated in textbook style, has frequently been used, and successfully at that, in the most diverse fields of mathematics, especially in set theory, by R. Dedekind, G. Cantor, F. Bernstein, A. Schoenflies, J. Kőnig, and others ⋯. Such an extensive use of a principle can be explained only by its self-evidence, which, of course, must not be confused with its provability. No matter, if this self-evidence is to a certain degree subjective – it is surely a necessary source of mathematical principles, even if it is not a tool of mathematical proofs".

26 Infinity Without Size 8.3 Rejecting the Axiom of Choosing

Zermelo's axiom of choosing should be rejected for these reasons:

1. Zermelo's attempt to justify his fictional axiom of choosing by comparing it with Euclid's parallel postulate, and by invoking the tenet that unprovability is not equivalent to nonvalidity, is inappropriate. The parallel postulate is reasonable, so it can be used as a basis for further theoretic development. But the axiom of choosing is not reasonable, and is fictional, as Zermelo admitted when he wrote in 1908 regarding the well ordering theorem, "Now in order to apply our theorem to arbitrary sets, we require only the additional assumption that a simultaneous choice of distinguished elements is in principle always possible for an arbitrary set of sets, or, to be more precise, the same consequences always hold as if such a choice were possible". The phrase "the same consequences always hold as if such a choice were possible" exposes his recognition that the axiom of choosing is fictional. Although it is true that unprovability is not equivalent to nonvalidity, this defense does not extend to unprovable vapidity.

Zermelo demonstrated the hollowness of the axiom of choosing in a letter in 1921 to the mathematics researcher Abraham Fraenkel. Zermelo wrote in regard to the axiom of choosing, 'There is no sense in which my theory deals with a real "choice".⋯ The "simultaneous choice" of elements and the gathering of these into a set ⋯ is for me only a way of envisaging matters which renders the (psychological) necessity of my axiom intuitive'.

2. In order to demonstrate the need to force the fictional axiom of choosing into the body of mathematical doctrine, Zermelo wrote in 1908 that there are seven theorems and problems which "could not be dealt with at all without the principle of choice ⋯".

Aside from the fact that an axiom should not be created if it is unreasonable and fictional, even if it is useful, five of the problems for which Zermelo said he needs the axiom of choosing are related to Cantor's theory of multiple infinities. Zermelo stated, "Cantor's theory of cardinalities, therefore, certainly requires our postulate ⋯". But Cantor's theory of multiple infinities is based on the mistaken idea of infinity as a completed set, rather than as a set of unending potential. So we conclude that Zermelo attempted to justify the creation of an unreasonable and fictional axiom by asserting that it is needed in order to resolve issues that arise within an invalid theory.

8.4 Inability to Well Order the Real Numbers

Zermelo used the axiom of choosing to prove Cantor's well ordering theorem. The axiom of choosing's choosing function automatically chooses one element from each of the infinite amount of subsets to be the first member of each of the respective subsets, and chooses an

27 Infinity Without Size element to be called "second", and chooses an element to be called "third", ⋯. All selections, both within each subset and across all subsets, take place simultaneously. If we take the union of all these well ordered subsets, we find that the set of real numbers has been well ordered.

As we have shown in Section 8.3, the axiom of choosing is invalid, so the proof for the well ordering theorem disappears.

It is useful to explore the well ordering of the set of natural numbers to get a clearer understanding of why the real numbers cannot be well ordered.

In the case of the set of natural numbers, the least element of any of the infinite amount of subsets is identified automatically and simultaneously. The set of natural numbers is the union of all subsets, which are all well ordered, so the set of natural numbers is well ordered.

When we say that the set of natural numbers is the union of all well ordered subsets, we mean:

1. All the subsets that the union is created from are subsets of the natural numbers, and these subsets are well ordered.

2. There is no subset of the set of natural numbers that is excluded, or that is not well ordered.

We now show why the well ordering present in the set of natural numbers does not apply in the case of the set of real numbers.

In the case of the set of real numbers, if we refer to the union of all well ordered subsets, we are intimating that:

1. All the subsets that the union is created from are subsets of the real numbers, and are well ordered.

2. There is no subset of the set of real numbers that is excluded, or that is not well ordered.

There is no way to automatically and simultaneously choose a first element of each subset. So the selection process would have to be done manually and arbitrarily. (The mythical axiom of choosing cannot be used, because the axiom is based on a fictional choosing function, so the ordering is artificial). An infinite amount of selections would require an infinite amount of time, so the process would not be able to be completed. It would only be possible to arbitrarily well order a union of a finite amount of subsets of the real numbers. So the requirement contained in part two of our definition of the union of all well ordered subsets of the real numbers would not be met, because there would be an infinite amount of subsets that would not be well ordered. Therefore, it is not correct to say that the set of real numbers can be well ordered.

28 Infinity Without Size 8.5 Creating Infinite Cartesian Product Using Axiom of Choosing

One of the areas that the axiom of choosing is used for is the cartesian product of sets 𝐴, 𝐵, 𝐶,, ⋯ where the expression 𝐴, 𝐵, 𝐶, ⋯ represents an infinite amount of disjoint finite unordered subsets of the infinite set of real numbers. The axiom of choosing is employed to assure that the cartesian product 𝐴𝐵𝐶⋯ can be calculated, just as we can calculate the cartesian product of a finite amount of sets 𝐴, 𝐵, 𝐶, as 𝐴𝐵𝐶. We will show why the proponents of the axiom of choosing consider the axiom of choosing essential, and sufficient, for creating the cartesian product.

Before we discuss the case of an infinite amount of finite sets, we first examine the case of a finite amount of finite sets. We can arbitrarily select a "first" element from each of the sets 𝐴, 𝐵, 𝐶 and combine these elements to form the first tuple of the cartesian product. Then we can arbitrarily select the "next" element from 𝐶, and combine the previously selected two 𝐴, 𝐵 elements with the new element from 𝐶. We continue making selections from the elements of 𝐴, 𝐵, 𝐶, until we arrive at a set which is the cartesian product 𝐴𝐵𝐶. At any particular point in the process of combining, we can make a variety of choices for "next". Each choice will determine a unique order for the cartesian product that we are in the process of forming. However, all versions of a cartesian product are equivalent, because order in a set is irrelevant.

However, if the amount of finite sets is infinite, we would need an infinite amount of time in order to make the required infinite amount of arbitrary selections, so the selection process can never be completed. Therefore, this type of selection is not viable.

We need a rule-based selection process that would allow us to instantly and simultaneously pick one member from each set and refer to it as first, and at the same time pick from each set the second, third, ⋯, and thereby order the sets. However, a set such as 𝑎,𝑎 does not have a first member because the set is unordered, and can also be written as 𝑎,𝑎.

The axiom of choosing solves this problem by providing an imaginary function which mechanically and automatically, in a deterministic and simultaneous manner, selects elements from each set, which we refer to as "first", "second", "third", ⋯, even though the members selected have no distinguishing characteristic that indicates that they should be chosen.

8.6 Infinite Cartesian Product Without Axiom of Choosing

The cartesian product of the infinite amount of subsets of the real numbers can be done algorithmically, and does not need the invalid axiom of choosing, as we will show.

We first examine the cartesian product of a finite amount of sets.

29 Infinity Without Size If 𝐴𝑎,𝑎,𝐵𝑏,𝑏,𝐶𝑐,𝑐, then

𝐴𝐵𝐶𝑎,𝑎 𝑏,𝑏 𝑐,𝑐

𝑎,𝑏,𝑐, 𝑎,𝑏,𝑐, 𝑎,𝑏,𝑐, 𝑎,𝑏,𝑐, 𝑎,𝑏,𝑐, 𝑎,𝑏,𝑐, 𝑎,𝑏,𝑐, 𝑎,𝑏,𝑐

We can also use a different method of combining the elements, which leads to this version of the cartesian product:

𝐴𝐵𝐶

𝑎,𝑏,𝑐, 𝑎,𝑏,𝑐, 𝑎,𝑏,𝑐, 𝑎,𝑏,𝑐, 𝑎,𝑏,𝑐, 𝑎,𝑏,𝑐, 𝑎,𝑏,𝑐, 𝑎,𝑏,𝑐

The order of the tuples in the two versions of the cartesian product is different. In the first version of the cartesian product, the element that experiences the first index change is the element of the last set (𝑐 becomes 𝑐), and the first changes for the first two sets occur later (𝑏 to 𝑏 and 𝑎 to 𝑎). In the second version of the cartesian product, the element that experiences the first index change is the first set (𝑎 becomes 𝑎), and the first changes for the last two sets occur later (𝑏 to 𝑏 and 𝑐 to 𝑐). However, the cartesian products are the same in both versions, because order in a set is irrelevant. Even though, for example, in the first version

𝑎,𝑏,𝑐 precedes 𝑎,𝑏,𝑐, and in the other version 𝑎,𝑏,𝑐 precedes 𝑎,𝑏,𝑐, the two cartesian products are the same.

The combining sequence of the first version of the cartesian product is standard, but we choose to use the second combining sequence, because it allows us to form a cartesian product of an infinite amount of sets. If we would use the first combining sequence, we would not be able to create a cartesian product, as we will show later.

We now write the cartesian product of three sets using a different notation than we used previously. We wish to obtain the cartesian product of the sets 𝑆,𝑆,𝑆, where each set has two elements 𝑠. In the notation 𝑠, 𝑖 is the index of the set, and 𝑗 is the index of the element within the set.

𝛱𝑆 𝑆 𝑆 𝑆 𝑠,𝑠 𝑠,𝑠 𝑠,𝑠

𝑠 ,𝑠 𝑠 , 𝑠 ,𝑠 𝑠 , 𝑠 ,𝑠 𝑠 , 𝑠 ,𝑠 𝑠 , , , , , 𝑠,𝑠,𝑠, 𝑠,𝑠,𝑠, 𝑠,𝑠,𝑠, 𝑠,𝑠,𝑠

30 Infinity Without Size We now consider the case where the domain of 𝑖 is 1, ∞, and the domain of 𝑗 is 1,2.

𝛱𝑆 𝑆 𝑆 𝑆 𝑆 𝑆 ⋯𝑠,𝑠 𝑠,𝑠 𝑠,𝑠 𝑠,𝑠 𝑠,𝑠 ⋯

= ⎧ 𝑠,𝑠,𝑠,𝑠,𝑠, ⋯ , 𝑠,𝑠,𝑠,𝑠,𝑠, ⋯ , 𝑠,𝑠,𝑠,𝑠,𝑠, ⋯ , 𝑠,𝑠,𝑠,𝑠,𝑠,⋯, ⎫ ⎪ 𝑠 ,𝑠 𝑠 ,𝑠 𝑠 , ⋯ , 𝑠 ,𝑠 𝑠 ,𝑠 𝑠 , ⋯ , 𝑠 ,𝑠 𝑠 ,𝑠 𝑠 , ⋯ , 𝑠 ,𝑠 𝑠 ,𝑠 𝑠 ,⋯, ⎪ , , , , , , , , ⎨ 𝑠 ,𝑠 𝑠 ,𝑠 𝑠 , ⋯ , 𝑠 ,𝑠 𝑠 ,𝑠 𝑠 , ⋯ , ⋯ ⎬ ⎪ , , , , ⎪ ⎩ ⎭

By using the subscripts 𝑠 and 𝑠, it may seem that we have ordered the two elements in 𝑆 prior to taking the cartesian product. However, the seeming designation of first and last is indeterminate, and also irrelevant. The designation is indeterminate because an infinite amount of arbitrary choices cannot be made, and irrelevant because the cartesian product is unaffected by the order in which the elements of 𝑆 appear in the set in preparation for forming the cartesian product.

Now we can explain why we cannot use the first combining sequence of the cartesian product of an infinite amount of sets. If the first combining sequence were used, the element whose index would change first is the element of the last set. But in a cartesian product of an infinite amount of sets, the last set is never reached, so the cartesian product would not be able to be created. All

tuples would uniformly appear as 𝑠,𝑠,𝑠,𝑠,𝑠,⋯. No matter how large we allow the value of 𝑖 in 𝑠 to become, all the tuples will be identical, so this formulation of tuples would not represent a cartesian product.

When we say that the cartesian product of an infinite amount of sets 𝑆 𝑆 𝑆 𝑆 𝑆 ⋯ is a set containing all tuples 𝑠,𝑠,𝑠,𝑠,𝑠, ⋯, we use this interpretation:

1. The cartesian product contains only tuples of the type 𝑠,𝑠,𝑠,𝑠,𝑠, ⋯.

2. There is no tuple of the type 𝑠,𝑠,𝑠,𝑠,𝑠, ⋯ that is excluded from the cartesian product.

An extension to any finite value of 𝑗 greater than 2 can be easily made.

We have discovered that in order to create the cartesian product of an infinite amount of sets, there is no need for the axiom of choosing, which provides an imaginary choosing function which, in an artificial sense, orders the elements in each set prior to taking the cartesian product.

31 Infinity Without Size 9.0 Misguided Reason for Creation of ZF Axiomatic Set Theory

The Zermelo-Fraenkel (ZF) axiomatic set theory was developed by Zermelo, and later modified by Fraenkel. Zermelo created his theory mainly in order to avoid Russell's Paradox (posed by the mathematics researcher Bertrand Russell) and Cantor's Paradox.

In the original theoretical framework for sets, a set is defined as a collection of elements, which are referred to as members of the set. Sets have operations such as union, intersection, negation, and cartesian product, relations such as equivalence, ordering, and function, and laws such as associative, commutative, distributive, and identity. In ZF set theory, the terms "set" and "member of" (denoted by the symbol ∈) are undefined, and axioms are used which serve as assumptions regarding the properties of sets and membership. Statements are derived from the axioms using the rules of inference provided by a logical system. In ZF, set theory is a logical abstraction which is not derived from common human experience regarding groups of items.

We will show that Russell's Paradox and Cantor's Paradox are invalid. Since these alleged problems with the original theoretical framework disappear, the basis for claiming that ZF set theory is needed in order to work in a reliable way with sets is defunct. The original, more direct framework for creating and using sets, inappropriately called "naive" set theory by the proponents of ZF set theory, does not lack any sophistication, and is useful, along with ZF set theory, in serving as a theoretical basis for understanding sets.

9.1 Russell's Paradox

If we have a set 𝐴sets that do not contain a reference to themselves, there is a paradox. If 𝐴 contains a reference to itself within its own set, then it should not contain that reference, and if it does not contain a reference to itself, then it should contain a reference to itself.

9.2 Refutation of Russell's Paradox

A set cannot reference itself because at the time of the formation of the set, which is when items are being entered, the set does not yet exist as a completed object.

In order for a reference to a set to qualify as an element in its own set, the decision on its qualification must be able to be made during set formation. However, we cannot determine whether the set should contain a reference to itself until the set formation is complete and we can evaluate the nature of the set to determine whether it meets the requirements necessary to be included in its own set. But by then it is too late to insert in the set a reference to itself. Therefore, the reference to the set is not eligible for inclusion. This lack of inclusion cannot be used post facto as a basis for asserting that there should be inclusion.

32 Infinity Without Size 9.3 Cantor's Paradox

The set of all sets 𝑆 has a cardinality |𝑆|, which would have to be the greatest possible cardinality that can exist. However, if we take the power set of 𝑆, 𝑃𝑆, we can obtain the cardinality |𝑃𝑆|, which is greater than |𝑆|, so |𝑆| is not the greatest possible cardinality.

9.4 Refutation of Cantor's Paradox

Although we demonstrated in Section 4.2 that a power set of an infinite amount of sets can be given a valid interpretation, neither the set of an infinite amount of sets, nor its power set, nor any set with an infinite amount of members, has a size. Since sets have no size, the basis for the paradox is eliminated.

10.0 The Multiverse Folly

The idea that infinity is a completed set leads to the nonsensical idea of a multiverse. A popular concept among a portion of physics researchers is that since the world is presumed to be infinite, we can adopt a view of the world as being composed of a multiverse, or an infinite amount of universes.

According to the multiverse theory, there is one universe where an accountant and a zoologist are sitting in a physics researcher's office, and he is explaining to them the theory of the multiverse by sketching a diagram on paper with his pencil, and there is a fly darting about in a particular zigzag fashion. There is another universe that is identical, except that the fly is moving in a slightly different zigzag manner. There is another identical universe, except that Napoleon and his officers are in an adjacent office, poring over maps and discussing military strategy. Another set of universes exist that are identical to these universes, except that the physics researcher is writing his proof with a pen instead of a pencil. The universes we have described are only an infinitesimal portion of the universes that exist in the world.

The universes we described do not consist of only a few office rooms, or a building, or a city. They are full-fledged copies of our universe, including all the galaxies, except that each universe has one or more differences in relation to the other universes. All possible variations must be accounted for, with a separate universe for each variation.

In addition, there are infinite identical copies of each universe.

According to the proponents of the multiverse, these universes are not fanciful flights of imagination, but exist by necessity, with all the features that we have described, since in an infinite world, every variation must exist.

33 Infinity Without Size The multiverse folly is caused by a misunderstanding of infinity. Infinity is the vastness of potential, but does not guarantee the presence of all things. The misconceived multiverse idea is based on the invalid principle that infinity means the inclusion of everything, so all possible variations must be regarded as existing.

We now describe a contradiction that arises in the multiverse concept.

According to the multiverse notion, there are infinite universes, with infinite variations. Then there is a certainty that there exists a universe where there are people who appear to be the same as us, but whose brains operate with different rules of logic. In one of these universes, there are physicists who justifiably conclude, based on a meticulous application of logical principles, that there is only one universe. Since this conclusion is irrefutable, it represents Truth. Truth is universal (or multiversal), so we must accept that conclusion. Since we know that our universe exists, then that only universe must be our universe, and there are no other universes. But if our universe is the only one, then our principles of logic represent Truth. Our principles of logic dictate that there is a multiverse. But if there is a multiverse, we conclude, based on the previous reasoning, that there is no multiverse. So we have a contradiction. If there is a multiverse, then there is no multiverse. If there is no multiverse, then there is a multiverse.

34 Infinity Without Size