PREFACE
PREFACE
Introduction
Welcome to "One to Infinity: A Guide to the Finite", a wholly remarkable book, and perhaps the first of its kind, quite likely the largest of it's kind, boasting over 3000 pages of content, rivaling the bible in sheer volume, and continuing to grow year after year with no end in sight.
This book attempts to tell the tale of the finite, from it's humble beginnings at the beginning of the number line, to some unfathomable "conclusion" where we may look back and say we "know" the contents of the infinite. This is not done in ignorance of the facts of the infinite: that the infinite is by definition inexhaustible. Rather this is done in full knowledge of the futility of the task, with the secondary aim to know the finite (and by extension the infinite) as thoroughly as humanly possible. To speak of numbers though we must do so with signs and sigils, for numbers are abstract entities admitting no form. As such this book is also the story of how numbers may be conceived of and represented and how we may use these representations to stride further and further into the number line; an exhaustive compendium of all existing human knowledge on large numbers, an exploration of the subject from a practioners stand point as well as an observers, a book, ultimately, about what is known, unknown, and what may never be known about large numbers.
This is such a lofty ambitious aim that I openly admit that I may never complete it, but it will not be for lack of trying. Hence I present this work not as some perfect complete edifice but rather as a "perpetual work in progress". I had decided early on in the inception of this web book that I would not wait for my great magnum opus to be complete in order to present it to the world, but rather I should present it as an ever evolving ever growing work of art and science, much like the subject it attempts to exhaust. It is however not an irony lost on me that my futile task is as inexhaustible as the numbers themselves, for we would never be done with counting all the numbers from one to infinity, and so is it not apropos that this book too should never cease to be written? It has been growing for 8 years and has went from a handful of articles to a massive tome on large numbers, and I have no intention of stopping now, I'm in it for the long haul.
I must point out in fairness that it is not the only book on the internet of its kind. After it's inception other similar books also emerged on the internet, perhaps most prominently another massive inscrutable tome called "Big Psi" with a similar mission, but also many much smaller imitators who modeled their books after my pioneering efforts. I would also be remiss to not point out that one of the earliest "texts" to deal rather seriously with the subject of large numbers for any extended period on the internet was Robert Munafo's famous "Numbers" and "Large Numbers" pages going back as far as 1996, well before the inception of my website. There was also Bowers' famous 2002 website which dealt with a large number system he was creating called "Array Notation" that served as partial inspiration for my own website. But in the case of Munafo's work he never claimed he was writing a book nor did he claim any special purpose other than to report all the information on large numbers he had seen discussed in books and websites. On the other hand Bowers' had little interest in discussing the history of large number notations and was primarily interested in pushing the theoretical limits of number systems through his own creation. My site was the first to fuse these two purposes together into a single volume: a book that was at once a compendium as well as a theoretical and metaphysical treatise, unifying the subject, expanding upon it, and explicitly codifying its purposes, motivations and philosophical outlook. I bring you the subject of large numbers not merely as a historical tour but also a tour of the numbers themselves, as understood and expressed through human symbolic abstraction.
But why work so tirelessly on such a mundane task of cataloging man's cornucopia of colorful ways of expressing and naming large numbers? Because the large numbers are themselves fascinating beasts, far larger than you'd ever imagine they'd get, far removed from the petty little known universe, ambling off into some great unknown abyss of abstraction of which the actual, the real, the known, shrinks to an infinitesimal point in the blackest and darkest of unknowns. It is my great privilege to have found these great dark abodes quite accidentally as a child exploring what the infinite truly contained and being quite surprised, exhilarated and simultaneously horrified at every development. And it turns out that I am hardly alone in this fascination. Indeed this hobby has existed probably as long as man has had opportunity to understand quantity and have means of expressing it, either verbally or in written form. It is this innate curiosity which impels me to be a channel through which numbers may speak. And I further believe that such a work is more than the sum of its numbers, for we are forced to admit a reality greater than our own, ever turned to the infinite reality, not of the real, but of the yet to be realized, the yet to be thought of. In short we taste that greater reality of which we may call the divine, and of which the manifest, the actual is but a vanishing quantity in the possible. If this great abyss has any draw for you then you already implicitly understand why I toil, but if you find yourself unmoved then there is no way I can explain it to you. If you are terrified of the prospect then turn away now: it only gets worse from here on out! But if fear is not enough to deter you than prepare to have your mind and your concept of infinity blown apart and expanded to accommodate heaven upon greater heaven ... repeatedly ...
Prerequisites:
What you need to Know to get started
So how much mathematical background is needed to approach this book with understanding? As part of my initial mission statement I had wanted to build the subject of large numbers from first principles and be completely self contained. While that is the ideal, in practice this book falls short of the mark. Much of the necessary background information needed to fully realize that aim is either still missing from the manuscript or poorly introduced. With time this situation should improve, in principle, since the web book is always being "improved". But admittedly when a conflict between presenting interesting new results and providing detailed explanations of old ones has come up it is usually more tempting to present the new ideas. For that reason it may yet be a long time until this book can really be called "self-contained".
Does this mean you need to be a professor of mathematics to understand this stuff? Absolutely not. I myself am but a student of mathematics, still in the process of learning myself. I can say with confidence that you don't need to know Calculus, analysis, topology,etc. or other college level mathematics to understand this site. We will have little use for what one might label "continuous mathematics" which involves irrational and fractional quantities, but instead we can confine ourselves (to the most part) completely with positive integral values. Large numbers is a largely discrete subject. This does not mean that we will need advanced number theory either. We can actually get by without knowing very much about the numbers at all, other than that there is a smallest positive integer and that each integer has a successor.
What you learn in a remedial college algebra course is probably enough to follow along. Certainly you should be familiar with decimal notation, the basic elementary operators along with exponents, scientific notation, and be at least conversant with concepts like variables, and functions. We will have very little need for the cartesian coordinate system, most of the functions growing so fast that graphing them with be of no use whatsoever. Some of the topics will cover more advanced topics like logarithms, but this is not essential to the main thrust of the book and so may be skipped without effecting later topics. The most important concepts here are actually those of computability theory, but these will be largely explained as computable and then uncomputable functions are introduced, so this is by no means prerequisite knowledge. Although it's difficult to gauge precisely the level of prerequisite knowledge for reading this book I would say that a high school education is probably sufficient. And don't worry if you've forgotten some of the math. A lot of this will be reviewed in the early chapters as we develop numbers literally from scratch.
On Organization of Existing Content
This book touches on many topics from mathematics, science, philosophy, and even day to day experience, but there is one underlying thread throughout all the content: we are only interested in other subject areas in so far as they involve large numbers and further our aim of generating larger and larger ones. This book attempts to touch upon everything that is even remotely related to large numbers so as to be as comprehensive as possible. But when a topic does come up, we will only need enough knowledge to appreciate the matter on numbers. It is by no means expected for the reader to be well versed in all scientific matters, and in fact it is not so much the science that will be of interest here so much as the numbers which can emerge from it. Given how potentially expansive the subject of large numbers is seeing as it can literally weave itself through almost any topic, some organization of the content, however imperfectly realized, is a necessity.
At the highest level of organization the book is broken up into a series of "Sections" that cover very broad developments in the science of large numbers. Currently there are only 4 sections available:
Section I deals with the introduction of the concept of number, the aims of our study of large numbers, and the development of numeration systems, both ancient and modern to cope with describing them, as well as a wealth of mathematical background to prepare the reader for the rest of the book
Section II introduces the reader to scientific notation, orders of magnitude, how large numbers can get in science, combinatorics, and other abstract studies, as well as exploring more extensive number-naming schemes designed to deal with all of these kinds of numbers. By the end of Section II the reader should have already been taken to about the limits of expressing large numbers by conventional means.
Section III introduces the reader to the basic mathematics of recursion theory, and examples of many popular large number notations are given and compared with numbers previously conceived in Sections I and II. More over it is demonstrated that mathematical recursion far outstrips any numbers in any practical application and far surpasses conventional number notations including decimal notation, scientific notation, exponents and factorials!
Section IV discusses much more esoteric and extensive systems which take recursion theory to its logical extreme by creating as complex and powerful a system as possible. Here I introduce my own massive system called Extensible-E
Future Sections will develop even further techniques for generating large numbers using theoretical constructions for defining very fast functions (many growing faster than large stretches of the recursive hierarchy), including the so called non-computable functions. After this there may be some theoretical discussion about the infinite and the infinite varieties of the infinite. The book will conclude with theoretical speculation as to the ever expanding future of studies into large finite and large infinite numbers and what we may never fully know about them in any future world. This is essentially the complete outline for the book.
The Sections are further broken down into chapters, which are further broken down into individual articles covering topics in a mostly modular way. This means that you are actually free to read a lot of the articles in any order you like, but the content is also written in a way that weaves a continuous narrative throughout the book so that you can read it from beginning to end as one developing story about numbers. If you'd like to read the book cover to cover you can begin by clicking the link below which will bring you to the first article of the book. From there every article contains a link at the top to the previous article and a link at the top to the next article, so that you can navigate the entire book from article to article in sequence to its end. If you would prefer to navigate to specific passages from a more lofty position you can also go back to the main hub and search the chapter hubs for content that catches your interest. I wish you a pleasant journey through the infinite worlds of the finite ...