Kurt Giidel maintained, and offered detailed proof, that in any arithmetic system, even in elementary parts of arithmetic, there are propositions which cannot be proved or disproved within the system. It is thus uncertain that the basic axioms of arithmetic will not give rise to contradictions. The repercussions of this discovery are still being felt and debated in 20th-century mathematics.The present volume reprints the first English translation of Giidel's far-reaching work. Not only does it make the argument more intelligible, but the introduction contributed by Professor R. B. Braithwaite (Cambridge University}, an excellent work of scholarship in its own right, illuminates it by paraphrasing the major part of the argument.This Dover edition thus makes widely available a superb edition of a classic work of original thought, one that will be of profound interest to mathematicians, logicians and anyone interested in the history of attempts to establish axioms that would provide a rigorous basis for all mathematics. Translated by B. Meltzer, University of Edinburgh. Preface. Introduction by R. B. Braithwaite.

    But what if free will doesn't exist? What if our lives are largely predetermined, hardwired in our brains - and our choices over what we eat, who we fall in love with, even what we believe are not real choices at all? Neuroscience is challenging everything we think we know about ourselves, revealing how we make decisions and form our own reality, unaware of the role of our unconscious minds. Did you know, for example, that:* You can carry anxieties and phobias across generations of your family?* Your genes and pleasure and reward receptors in your brain will determine how much you eat?* We can sniff out ideal partners with genes that give our offspring the best chance of survival?Leading neuroscientist Hannah Critchlow draws vividly from everyday life and other experts in their field to show the extraordinary potential, as well as dangers, which come with being able to predict our likely futures - and looking at how we can alter what's in store for us. Lucid, illuminating, awe-inspiring The Science of Fate revolutionises our understanding of who we are - and empowers us to help shape a better future for ourselves and the wider world.

    this is one of those rare books that will change your beliefs - and in doing so will change your life. tHE ANt AND tHE FERRARI offers readers a clear, navigable path through the big questions that confront us all today. What is the meaning of life? Can we be ethical beings in today's world? Can we know if there is life after death? Is there such a thing as Absolute truth? What caused the Big Bang and why should you care?

    Penrose, a cat with a knack for math, takes children on an adventurous tour of mathematical concepts from fractals to infinity.

    A timeless introduction to the field and a landmark in symbolic logic, showing that classical logic can be treated algebraically.

    Aimed at undergraduate students in mathematics, physics, and engineering, the book's intuitive explanations, lack ofadvanced prerequisites, and consciously user-friendly prose style will help students to master the subject more readily than was previously possible. The key to this is the book's use of new geometric arguments in place of the standard calculational ones. These geometric arguments are communicatedwith the aid of hundreds of diagrams of a standard seldom encountered in mathematical works. A new approach to a classical topic, this work will be of interest to students in mathematics, physics, and engineering, as well as to professionals in these fields.

    This concept is at the root of the computational worldview, which basically says that very complex systems — the world we live in — have their beginnings in simple mathematical equations. We've lately come to understand that such an algorithm is only the start of a never-ending story — the real action occurs in the unfolding consequences of the rules. The chip-in-a-box computers so popular in our time have acted as a kind of microscope, letting us see into the secret machinery of the world. In Lifebox, Rucker uses whimsical drawings, fables, and humor to demonstrate that everything is a computation — that thoughts, computations, and physical processes are all the same. Rucker discusses the linguistic and computational advances that make this kind of "digital philosophy" possible, and explains how, like every great new principle, the computational world view contains the seeds of a next step.

    Army and Navy and the British government over the last five years. Now, Battle of Wits presents the history of the war that these documents reveal. From the battle of Midway until the last German code was broken in January 1945, this is an astonishing epic of a war that was won not simply by brute strength but also by reading the enemy's intentions. The revelations of Stephen Budiansky's dramatic history include how Britain tried to manipulate the American codebreakers and monopolize German Enigma code communications; the first detailed published explanations of how the Japanese codes were broken; and how the American codebreaking machines worked to crack the Japanese, the German, and even the Russian diplomatic codes. The compelling narrative shows the crucial effect codebreaking had on the battlefields by explaining the urgency of stopping the wolf pack U-boat attacks in the North Atlantic, the importance of halting Rommel's tanks in North Africa, and the necessity of ensuring that the Germans believed the Allies' audacious deception and cover plans for D-Day. Unveiled for the first time, the complete story of codebreaking in World War II has now been told.

    The material of this book is indispensable for anyone who wishes to understand the real spiritual roots of Western civilization.

    The year 2000 is alternatively the year 2544 (Buddhist), 6236 (Ancient Egyptian), 5761 (Jewish) or simply the Year of the Dragon (Chinese). The story of the creation of the Western calendar, which is related in this book, is a story of emperors and popes, mathematicians and monks, and the growth of scientific calculation to the point where, bizarrely, our measurement of time by atomic pulses is now more accurate than time itself: the Earth is an elderly lady and slightly eccentric - she loses half a second a century. Days have been invented (Julius Caesar needed an extra 80 days in 46BC), lost (Pope Gregory XIII ditched ten days in 1582) and moved (because Julius Caesar had 31 in his month, Augustus determined that he should have the same, so he pinched one from February).

    At 70,000 words and 450 pages, The Anxious Truth goes well beyond just telling that that you'll get better. It actually shows you exactly how! You're anxious all the time, experiencing panic attacks over and over, and maybe afraid to leave your house or to be left alone for even a few minutes. You are avoiding simple things like driving, eating in restaurants, attending family functions, or going to the supermarket. You are terrified of the next wave of anxiety or the next panic attack. Your anxiety problems are ruining your relationships, your family life, and your career. Your anxiety problems have you afraid, confused, lost, and feeling hopeless. How did you get here? What went wrong? You've tried so many things, but nothing has cured your anxiety? What can you do now?This book, written by a former anxiety sufferer, best-selling author of "An Anxiety Story", and host of The Anxious Truth podcast will walk you through exactly how you got to where you are today, why you are not broken or ill, and what the true nature of your anxiety disorder is. Next, the book will walk you through what it takes to solve your anxiety problems, how to make an anxiety recovery plan, then how to correctly execute that plan.The Anxious Truth isn't always what you want to hear, but it's what you NEED to hear in order to solve this problem once and for all and move toward the life you so desperately want. Based firmly on the principles of cognitive behavioral therapies that have been shown over decades to be most effective in treating anxiety problems, the Anxious Truth will teach you how to move past your anxiety symptoms, past endless digging for hidden "root causes", and into an action oriented plan that will help your brain un-learn the bad reaction and fear habits that have gotten you into this predicament. The Anxious Truth will take the cognitive mechanism that got you into a corner, throw it in reverse, and use it to your advantage, backing you out of this jam and into a life free from irrational fear and needless avoidance.More than just a book, The Anxious Truth goes hand-in-hand with The Anxious Truth podcast (theanxioustruth.com) and the growing and vibrant social media community surrounding it. Read the book, listen to five years worth of free podcasts chock full of helpful advice and information, and join a large online community of fellow anxiety sufferers that are done talking about this problem and ready to actually take action to solve it. Change is possible. No matter how long you've suffered with your anxiety issues, you can get better. The Anxious Truth will tell you what you need to hear and will arm you with the information, understanding, and skills you need to get the job done.Let's do this together!

    Each chapter includes many illustrative examples to assist the reader. The book emphasizes methods for finding solutions to differential equations. It provides many abundant exercises, applications, and solved examples with careful attention given to readability. Elementary Differential Equations includes a thorough treatment of power series techniques. In addition, the book presents a classical treatment of several physical problems to show how Fourier series become involved in the solution of those problems. The eighth edition of Elementary Differential Equations has been revised to include a new supplement in many chapters that provides suggestions and exercises for using a computer to assist in the understanding of the material in the chapter. It also now provides an introduction to the phase plane and to different types of phase portraits. A valuable reference book for readers interested in exploring the technological and other applications of differential equations.

    The Greatest Secret of All clearly explains this law of manifestation but then takes it a quantum leap further, revealing what is truly important in life. We have what we need within us to do what we love, to be the people we dream of being, and to become completely fulfilled along the way — to become, as Abraham Maslow put it, self-actualized. We also have the capability, here and now, to create a world that works for everyone. In these pages, you will find the secret to a life of happiness, inner peace, ease, and fulfillment, and the secret that lets each of us contribute to making the world a better place for all.

    There is a kind of religion that is lifeless and is the bitter enemy of true faith. Think of Dietrich Bonhoeffer’s prophetic attempts to waken the sleeping German church, often unwitting allies of Hitler and the Third Reich. Or of William Wilberforce’s heroic efforts to rouse his complacent “Christian” countrymen to stand against the monstrous evil of the slave trade. Bonhoeffer and Wilberforce stood against the evil of their times—an evil often repackaged in religious-sounding language.Eric Metaxas’s electrifying message—delivered before the president and dozens of national leaders at the Sixtieth Annual National Prayer Breakfast—calls readers to follow in the steps of Wilberforce and Bonhoeffer, men who lived their faith and swam against the mainstream, instead of drifting along with it. Metaxas makes it clear that phony religiosity offends God himself—and that real prayer is only possible with a living faith in a living God. And that kind of faith can transform the world. No pressure.

    An Introduction to Game Theory, by Martin J. Osborne, presents the main principles of game theory and shows how they can be used to understand economic, social, political, and biological phenomena. The book introduces in an accessible manner the main ideas behind the theory rather than their mathematical expression. All concepts are defined precisely, and logical reasoning is used throughout. The book requires an understanding of basic mathematics but assumes no specific knowledge of economics, political science, or other social or behavioral sciences. Coverage includes the fundamental concepts of strategic games, extensive games with perfect information, and coalitional games; the more advanced subjects of Bayesian games and extensive games with imperfect information; and the topics of repeated games, bargaining theory, evolutionary equilibrium, rationalizability, and maxminimization. The book offers a wide variety of illustrations from the social and behavioral sciences and more than 280 exercises. Each topic features examples that highlight theoretical points and illustrations that demonstrate how the theory may be used. Explaining the key concepts of game theory as simply as possible while maintaining complete precision, An Introduction to Game Theory is ideal for undergraduate and introductory graduate courses in game theory.