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A Review of Infinite Numbers and the Convergence of Divergent

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A Review of Infinite Numbers and the Convergence of Divergent Sequences. David Alan Paterson CSIRO CMIS Graham Rd, Highett, 3090 Australia January 23, 2012 Contents I Preliminaries 4 1 Introduction 5 1.1 Introduction . 5 1.2 Notation . 8 2 How mathematicians treat infinity 11 2.1 Infinity, infinity, WHICH infinity? . 11 3 Commutative ordinals, self-commutative functions and Big Θ notation 21 3.1 Commutative ordinals . 21 3.2 Self-commutative functions . 23 3.3 Big Θ notation . 24 II Du Bois-Reymond's pantachie 27 4 Du Bois-Reymond 28 4.1 Hardy's exposition from 1910 . 28 4.2 Transcending the LE scale . 38 5 Completion of the du Bois-Reymond pantachie 44 5.1 Borel's ladder of types of increases . 44 5.2 Completion of the Pantachie . 50 5.3 On the pantachie, what is a number? . 52 III Limits 54 6 Archimedean classes 55 6.1 Archimedean axiom, Archimedean class, and prototype . 55 6.2 Paterson's attempt to enumerate the Archimedean classes . 56 1 7 Philosophy of limit 62 7.1 A serious inconsistency . 62 7.2 Overcoming the inconsistency . 63 8 Limit of smooth & regular functions 66 9 Rational function of leading term limits 68 10 Sequences and functions with non-regular growth 73 11 Limits of sequences, some traps to avoid 83 IV Surreal numbers as limits 85 12 Surreal numbers 86 12.1 Introduction to surreal numbers . 86 12.2 NBG set theory . 91 12.3 Standard and nonstandard definitions . 93 12.4 Operations on the surreals at generation n! . 96 13 Surreal numbers as limits 98 13.1 Real numbers as limits of sequences of dyadic rationals . 98 13.2 The du Bois-Reymond calculus of infinities and the surreal numbers . 100 13.3 Nomenclature comparison . 101 13.4 Rational infinite numbers as a subset of surreal numbers . 102 13.5 The gap between In and NO . 105 13.6 The du Bois-Reymond calculus of infinity and the surreal num- bers . 108 V Other infinite number systems 109 14 Hahn series and other series 110 14.1 Laurent series . 110 14.2 The Levi-Civita field . 110 14.3 Transseries . 111 14.4 Hahn series . 113 2 15 Hyperreal numbers 115 15.1 The axiomatic approach . 115 15.2 The ultrafilter approach . 119 16 The non-existence of sin ! 123 17 Veronese-Hilbert continuum 126 VI Applications 131 18 Classical paradoxes resolved 132 18.1 Ground rules . 132 18.2 Paradoxes involving the summation of series . 133 18.3 Classical and other paradoxes . 140 18.4 Achilles and the Tortoise . 146 19 More Series 150 20 Limits on real and complex numbers 152 21 Numerical integration 155 22 Fourier's method for general functions 158 VII Conclusions 161 23 Summary of progress 162 24 Conclusions 164 3 Part I Preliminaries 4 Chapter 1 Introduction 1.1 Introduction Difficult mathematical problems usually involve infinity in some form. Pure mathematics requires extreme care in defining exactly on which particular domain each mathematical technique is defined. Imagine a mathematical utopia in which infinity is as easy to handle as the number 2, and in which extreme care is not as much of a problem because sequences always converge. Thinking about this led me to ask the crucial \what if " question. \What are the direct consequences of the assumption that limn!1 n 6= limn!1 2n?". When I asked this question, I had no idea what amazing mathematical vistas it would lead to. \What if " questions have an important role in the history in mathematics. The most famous is the question \What if Euler's fifth postulate doesn't hold?". Euler's fifth postulate is that \through any point it is possible to draw exactly one line parallel to a given line". Rejection of Euler's fifth postulate led to the non-Eulerian geometries - Riemannian spherical geometry and Lobschevsky hyperbolic geometry. Less well known is the \what if " question, \what if the Archimedean axiom doesn't hold?". There are several ways to express the Archimedean axiom, one is that \there is no number larger than the integers". Rejec- tion of the Archimedean axiom in 1891 by Veronese led to non-Archimedean geometry. The concept of \non-Archimedean", that there exists a number larger than all integers, turns out to be an ideal definition of infinity. Rather than 5 using the symbol 1 to write this number, mathematicans use !. I find this doubly appropiate because not only is ! the last letter of the Greek alphabet, it also looks like 1 with the top chopped off. Once we choose an ! we are free to operate on it with normal operations such as addition, multiplication, taking it to a power, taking the logarithm of it, etc. The result is an ordered field of infinite, finite and infinitesimal numbers that is directly analogous to, but is much larger than, the field of real numbers. Call it the ordered field of hyperreal numbers, or \hyperreals" for short, and write it ∗R to show that it is an extension of the ordered field of real numbers R. It turns out that non-Archimedean number systems tie in with very closely with my question \limn!1 n 6= limn!1 2n?". To make sense of this question it became necessary to devise a new definition of limit, an asymp- totic limit. In school we are taught two definitions of limit, the limit of a function at a real (or complex) number, and the limit of a sequence at infinity which is known to mathematicans as the (, δ) limit. When extended to the field of hyperreals, the two types of limits turn out to be incompatible in the sense that they always give different answers when the limit of a sequence is an infinitesimal number. To overcome this incompatibility, I devised an asymptotic limit of se- quences that turns out to be a generalization of a type of limit called Borel summation. There are already many generalizations of Borel summation, but this one seems to be new. So far as I can tell, it is sufficiently powerful that it will allow you to find the limit of every sequence, no matter how rapidly it \diverges", no matter how wildly it oscillates, and no matter how chaotically or randomly it fluctuates. In fact it's so powerful that it may end up relegating the very concept of \divergence" to the wastebasket of history. This new type of asymptotic limit works by first mapping a general se- quence onto one that has no oscillations or random or chaotic fluctuations by using a technique called the \ensemble mean" borrowed from time series analysis. Each sequence, limn!! f(n) is split into the sum of smooth s, peri- odic p (including Lebesgue integrable), chaotic k and random r components 1 P j j P k k P l as follows: f(n) = s(n) + j s(n) p(n) + k s(n) k(n) + l r(n) where each appearance of s can be a different smooth sequence. The ensemble mean of each of these components, or the mean of the mean, will usually be a smooth sequence. Care is taken to ensure that each mean is definied locally at a sufficiently large n rather than being smeared over several values of n. At this stage the sequence may still \diverge" rapidly, but what I'm calling \smooth" has what mathematicans call \regular increase". That allows the second stage of the limit operation to map the result onto what is either a subset of Conway's ordered field of surreal numbers or the entire surreal numbers. Recent work by Ehrlich has found a one to one mapping between 6 the surreal numbers and the hyperreal numbers, so it all ties together. The concept of \regular increase" comes from a third approach that math- ematicans have had to the ordered field of infinite, finite and infinitesimal numbers called the du Bois-Reymond infinitary calculus. This looks at the ratios of functions f(x)/φ(x) as x tends to infinity, and assigns functions f and φ to the same Archimedean class when this ratio tends to a nonzero finite number. The limits of sequences can be absorbed into the du Bois-Reymond infinitary calculus because a sequence is simply defined as a function sampled at the natural numbers. Work on the du Bois-Reymond infinitary calculus has been held up somewhat by the problem of handing functions that do not have a \regular increase", but the new type of asymptotic limit overcomes that problem. So it looks like the du Bois-Reymond infinitary calculus will eventually be able to sit beside the surreal numbers and the hyperreal num- bers as an alternative definition of the complete ordered field of infinite, finite and infinitesimal numbers. The ordered field of real numbers can be defined in one of four ways: ax- iomatically, as limits of sequences, as Dedekind cuts, and from the geometry of the continuum. It now looks possible that the complete ordered field of infinite, finite and infinitesimal numbers can be defined in up to nine different ways. Four are modifications of the definitions of real numbers. These are axiomatically (one approach to the hyperreals), as limits of sequences (the present work), as cuts (Conway's surreal numbers), and from the non-Archimedean geometry of the continuum. The other five are as ultrafilters on sequences (the other approach to the hyperreals), as ratios of functions (du Bois-Reymond), as Big Θ notation, as Hahn sequences, and through infinitesimals. In researching this monograph, several rediscoveries were made.
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