On the Structure and Sizes of Infinitely Large Sets of Numbers

Transcription

1 1 On the Structure and Sizes of Infinitely Large Sets of Numbers Introduction: This paper introduces The Axiom for the existence of finite integers. The Axiom shows that sets of integers having only finite integers cannot possibly satisfy the current definition for infinite sets. In the paper, sets satisfying the current definition are called self-inclusive infinite sets. The concepts of largest finite integer and smallest infinite integer are introduced. They refine our understanding of what constitutes a self-inclusive infinite set of integers. They also lead to a new and more fundamental definition for infinite sets that is consistent with the current definition but permits infinite sets that are not self-inclusive. The Natural Numbers set, the Real Numbers set and its subsets, are finite number sets of the not self-inclusive kind. Their sizing is enabled via the following new definitions: The largest finite integer Q ; The smallest infinite integer N! ; Finite integer ; Infinitely large set of integers ; and Infinitely large set of elements. The Current Definition for Infinitely Large Sets (i.e., for Self-Inclusive Infinities): A set is infinitely large if there s a one-to-one correspondence between the set s elements and elements of the set s subsets. The paper calls such sets self-inclusive infinite sets because the correspondences exist between the full set and its internal subsets. Our Understanding of the Current Definition is Incomplete: The depiction S(I) = [1, 2, 3, 4, ] is understood as being a set of finite integers that satisfies the current definition. The continuation symbol conveys that the set is open-ended and has no ending so is infinitely long. However, there s detailed information to be had that s hiding in that portion of the set marked. We begin to reveal those details and the set s true structure by exploring how integers come into being, as follows. Creating the Integers: Integers are artificially created from our imagination as part of a counting process. This perspective was shared by Richard Dedekind i who considered counting itself as nothing else than the successive creation of an infinite series of positive integers in which each individual is defined by the one immediately preceding. Inherent in Dedekind s creative process is that integers exist only after they have once been named. This understanding leads to an axiom that is defined below using language that limits Dedekind s counting to a finite series of finite positive integers. The finite limitation is necessary because his an infinite series of positive integers is seriously problematical, as will soon be seen when we create the positive set of finite integers via the naming process. Definition. The axiom for the existence of finite integers: Finite integers exist only after they have once been named. To illustrate the axiom, let us imagine naming, for the first-time-ever, the finite integers in the set S(Q) = [1, 2, 3,... Q]. Since integers larger than Q have yet to be

2 2 named, they do not yet exist. This fact will be true no matter how large Q is. This truth must be preserved when developing concepts of infinity. The Current Definition for Infinitely Large Sets Fails for S(Q): The current definition in effect requires every finite integer in S(Q) to be at the beginning of at least one infinitely long subset of increasingly larger finite integers. At no time can this requirement be satisfied since subset integers are constrained from being larger than the finite integer Q, with the result that subsets will be finite in length. To illustrate: the subset that begins with (Q-1) ends in Q and cannot ever contain more than 2 finite integers regardless of how large Q is allowed to become enroute to infinity. The inability of the subsets to ever be infinitely large means S(Q) can never satisfy the current definition. How to Preserve the Current Definition: The challenge then is to determine how to preserve the set of positive integers as being a self-inclusive infinite set that involves S(Q). The proposed solution is to assume the existence of infinite integers, but such an entity has yet to be defined in mathematics. Let us proceed toward a definition for infinite integer. The Infinite Integer (The Concept): We return to the sequence [1, 2, 3,... Q], but where Q is the largest finite integer that we are able to name (count to) at some extremely distant and undefined future instant t in time. At each such instant or moment t there is, waiting to be named, an integer N that is one unit greater than Q. At any given time in the future there will always be such an N. N s value can never be written down because doing so requires a new instant in time which results only in the expressing of a new Q. In this way, N is inexpressibly (unattainably) large in a finite sense, which is the overarching defining attribute of being infinitely large. Creating Infinite Integers: Q symbolizes the largest finite positive integer and N symbolizes the smallest infinitely large positive integer. When we named and created the finite integer Q we also created N. N is the first positive whole number of the infinite realm the realm in which no finite numbers can occur. N can be renamed N! when desirable to emphasize that it is an infinite integer. Let us now allow S(Q) to include N. From this point onward in the paper, S(Q) depicts the set of positive finite integers going to infinity. S(Q) = [0, 1, 2, 3,... N]. All integers of the infinite realm can be listed as shown below, using S(Q) as a pattern. S(N) = [N, N+1, N+2, N+Q] S(2N) = [2N, 2N+1, 2N+2, 2N+Q] S(3N) = [3N, 3N+1, 3N+2 3N+Q]...

3 3 Fully Understanding the Self-Inclusive Infinite Set of Integers: The infinite set of integers, previously shown as (SI) = [1, 2, 3, 4, ], can now be shown as S(I) = [S(Q), S(N), S(2N), S(3N), ]. As shown, S(I) contains all of the positive finite and positive infinite integers that are possible. Every finite integer, even (Q-1), is the beginning of an unending set of integers that s comprised mainly of infinite integers. Such sets, along with sets comprised entirely of infinite integers, are the subsets of S(I). They enable S(I) to be a self-inclusive infinite set that satisfies the current definition. It was observed earlier that N is inexpressibly (unattainably) large in a finite sense, which is the overarching defining attribute of being infinitely large. That observation derives from the infinite integers concept and forms the basis of a new definition that is consistent with the current definition, but is more fundamental. The new definition opens our eyes to the existence of infinitely large sets that are not self-inclusive infinite sets. The formalized version of the new definition is given below along with prerequisite other new definitions. Integers and Infinite Sets (New Definitions): The infinite integers concept enables the following new definitions. Definition. The largest finite integer Q: Q is imagined to be the largest finite whole number that can ever be expressed in answer to the question What is the largest whole number that can be named at this moment in time? Definition. The smallest infinite integer N! : The smallest infinite integer, N!, is an inexpressible number that is one unit larger than Q, where Q is imagined to be the largest finite whole number that can ever be expressed in answer to the question What is the largest whole number that can be named at this moment in time? Definition. Finite Integer: A finite integer is any whole number (including zero) that lies between the smallest plus and minus infinite integers, N! and (- N! ). Definition. Infinite set of integers: An infinite set of integers contains at least N! integers where N!, is an inexpressible number that is one unit larger than Q and where Q is imagined to be the largest finite whole number that can ever be expressed in answer to the question What is the largest whole number that can be named at this moment in time? Definition. Infinite set of elements: An infinite set of elements is one that contains at least N! elements, where N! is an inexpressible number that is one unit larger than Q and where Q is imagined to be the largest finite whole number that can ever be expressed in answer to the question What is the largest whole number that can be named at this moment in time? Comparing Infinite Sets: The preceding definitions lead us to understand comparisons of numerical infinities differently. To illustrate, let s consider comparing the set of positive integers to its subset

4 4 of positive even integers. The comparison can now be made in a manner that shows the integers Q and N! and in the context of the finite and infinite realms, as in Figure 1 where Q is assumed to be even. Figure 1. Comparison of Numerical Infinities...Finite realm.... Infinite realm Q/2 (Q/2+1) (Q/2+2) Q N! Q (N! +2) (N! +4) (2 N! -2) (2 N! )...Finite realm.... Infinite realm We have always understood such comparisons to be between infinitely long finite number sets. However the preponderance of correspondences involve infinite integers. We have always misunderstood such comparisons because we did not recognize and apply the Axiom for the existence of finite integers. Sizing the Infinities of the Finite Number System: The Infinite Set of Natural Numbers: The set of natural numbers is comprised of finite integers 1 to Q. Because the numbers Q and N! have been established, we are now able to show the set of natural numbers going to infinity in the closed-ended form of Figure 2. The set necessarily includes N! in order to satisfy the go to infinity requirement. The closed-ended form is an infinity that is N! in size. Figure 2. The infinity of natural numbers 1,2,3,4,5,... (Q-1), Q, N!. The Infinite Set of Real Numbers: The set of real numbers is comprised of the finite integers (going to infinity), the periodic decimal fractions (the rationals, obtained by division of finite integers), and the non-periodic decimal fractions (the irrationals, i.e., those decimal fractions that are not rationals). The Infinity of Finite Integers that Go to Infinity: The infinitely large set of integers that runs to +/- N! from zero is shown in Figure 3. Figure 3. The infinity of integers -N!,-Q,-(Q-1)... -5,-4,-3,-2,-1, 0, 1,2,3,4,5,... (Q-1), Q, N!. Figure 3 s infinitely large set has 2N! +1 numbers when zero is included. For simplicity s sake, we can say that the infinity of integers (going to infinity being understood) equals 2N! since 2N! +1 is not significantly different from 2N!. It is meaningful to note that numbers such as Q, (Q-1), and their equivalents (N! -1), (N! - 2), are not just finite, they are intangibly finite. The Rationals: For fractions between consecutive integers, we consider sequences that produce unique rationals smaller than unity. In this paper, unique rationals are those that are produced the first time in our considerations. Those that come later that duplicate

5 5 the decimal expression of the first are not unique. For instance, if 1/2=0.5 is encountered first, 10/20=0.5 is not unique. The first sequence of unique rationals is where 1 is the numerator. It is [1/2, 1/3,1/4, 1/5, 1/6, 1/Q,] and there are (Q-1) unique rationals. It takes only two unique rational fractions from the sequence where 2 is the numerator, e.g., (2/3, 2/5), to bring the number of identified unique rationals to N!. There will be many other unique rationals where the numerator ranges from 2 to Q. Each of those numerators will produce fewer than Q unique rationals, hence collectively they will produce fewer than Q 2 (ergo, fewer than N! 2 ) rationals. Thus, between consecutive integers there is an infinity of rationals that is greater than N! but less than (N! +N! 2 ). Since the Real Number Set contains 2N! integers, the Rational s set contains between 2 N! 2 and (2N! 2 +2N! 3 ) rationals, or, simplifying, between 2 N! 2 and 2N! 3 rationals. Half of these are positive and half negative. The Irrationals: Let us agree, as commonly understood, that there is an infinity of irrational fractions between consecutive integers. Assume there will be at least one infinitely large subset, called SS1 (subset1), that contains only N! such irrationals. It is known that multiplying an irrational number by a unique rational creates a new irrational. It is assumed such new irrationals are always unique irrationals, never duplicates. We know that [2/3, 2/5, 1/2, 1/3,1/4, 1/5, 1/6, 1/Q,] is a set that has N! unique rationals that are less than unity. Let s call that set the base set. If the N! unique numbers of the base set are applied (multiplied) against every irrational number in SS1, doing so will create (N! ) 2 new irrational numbers. Applying the base numbers to those new numbers creates (N! ) 3 additional new irrationals. Applying the base s numbers repeatedly for X times will produce (N! ) (X+1) additional new irrationals. Base numbers can be applied repeatedly for N! times, generating over (N! ) N! additional new irrationals. Even after (N! ) N! new irrationals have been generated, the reality is that more still can be generated by repeating the process. Note that no attempt has been made to sum the number of new irrationals so far produced along the way. The paper arbitrarily chooses to use the indeterminate term (N! ) >N! as the sum of the preceding unending proliferation of unique irrationals, and as the size of the Irrational s infinity. Given that there are 2N! integers in the Real Number Set, it can now be said that the Real Number Set contains 2N! (N! >N! )=2(N! ) (>N!+1) irrational fractions, half of which are positive and half negative. The size of the Real Number Set, now that its components have been sized, is R! =[2N! integers + between (2N! 2 & 2N! 3 ) rational fractions+ 2(N! ) (>N!+1) irrational fractions]. Since 2N!, 2N! 2, and 2N! 3 are all negligible relative to 2(N! ) (>N!+1), and since 2(N! ) (>N!+1) is insignificantly different from 2N! >N!, the expression for R! simplifies to R! =2N! >N!. Summary Table of Number Set Infinities: Numerical Infinities as Derived from Finite Integers Number Set Size of Set Order of Infinity Real Numbers >N! R! =2N! >N!

6 6 Irrationals >N! Rationals (between 2N 2! & 2N 3! ) 2, if not 3 Integers 2N! 1 Naturals N! 1 2N! >N! Author s declaration: This paper considerably modifies earlier thoughts about infinitely large numbers and infinite sets presented in a book that was self-published in 2004 [ ii ]. i Stephen Hawking, GOD created the Integers, Running Press Book Publishers, page 1079 [ ii ] Terry Mandzy, P. Eng., Infinity, God?, & Relativity Theory (Intangibles of Math & Physics), Epic Press, Belleville (Ontario, Canada), 2004, pages 29-53