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Author Stillwell, John

Title Roads to infinity : the mathematics of truth and proof / John Stillwell

Publ Info Natick, Mass. : A K Peters, [2010]
 PC  QA248 .S778 2010    AVAILABLE
 RWU Main Library  QA248 .S778 2010    AVAILABLE
 Salve Main Collection  QA248 .S778 2010    AVAILABLE
Descript xi, 203 pages : illustrations ; 24 cm
text rdacontent
unmediated rdamedia
volume rdacarrier
Note Includes bibliographical references and index
Contents Preface -- 1. The Diagonal Argument. -- Counting and Countability -- Does One Infinite Size Fit All? -- Cantor's Diagonal Argument -- Transcendental Numbers -- Other Uncountability Proofs -- Rates of Growth -- The Cardinality of the Continuum -- Historical Background -- 2. Ordinals. -- Counting Past Infinity -- The Countable Ordinals -- The Axiom of Choice -- The Continuum Hypothesis -- Induction -- Cantor Normal Form -- Goodstein's Theorem -- Hercules and the Hydra -- Historical Background -- 3. Computability and Proof. -- Formal Systems -- Post's Approach to Incompleteness -- Gödel's First Incompleteness Theorem -- Gödel's Second Incompleteness Theorem -- Formalization of Computability -- The Halting Problem -- The Entscheidungsproblem -- Historical Background -- 4. Logic. -- Propositional Logic -- A Classical System -- A Cut-Free System for Propositional Logic -- Happy Endings -- Predicate Logic -- Completeness, Consistency, Happy Endings -- Historical Background -- 5. Arithmetic. -- How Might We Prove Consistency? -- Formal Arithmetic -- The Systems PA and PA -- Embedding PA and PA-- Cut Elimination in PA -- The Height of This Great Argument -- Roads to Infinity -- Historical Background -- 6. Natural Unprovable Sentences. -- A Generalized Goodstein Theorem -- Countable Ordinals via Natural Numbers -- From Generalized and Ordinary Goodstein -- Provably Computable Functions -- Complete Disorder Is Impossible -- The Hardest Theorem in Graph Theory -- Historical Background -- 7. Axioms of Infinity. -- Set Theory without Infinity -- Inaccessible Cardinals -- The Axiom of Determinacy -- Largeness Axioms for Arithmetic -- Large Cardinals and Finite Mathematics -- Historical Background
Note "This sequel to the author's Yearning for the Impossible provides a readable survey of logicians' efforts to explicate the notions of truth, proof and undecidability, including the quest to find examples of atural' undecidable statements. Leavened with historical details, it focuses on the role of infinitary considerations in the development of modern mathematics, with particular attention to the undervalued contributions of Emil Post and Gerhard Gentzen."---John W. Dawson, Jr., author of Logical Dilemmas: The Life and Work of Kurt Godel --
"Stillwell has provided an accessible, scholarly treatment of all the foundational studies today's well-rounded professional mathematician ought to know, and has managed to do so in just over 200 pages. And that includes all the relevant history. I highly recommend it."---Keith Devlin, Stanford University, author of The Millennium Problems and co-author of The Computer as Crucible: An Introduction to Experimental Mathematics --
"Professor Stillwell ... succeeds, in every topic treated, in bringing a fresh eye to questions even mathematicians might think have been mined in the past to boring exhaustion [and] shows there is still a lot of gold to be found, if one only thinks about things in a new way. Stillwell brings new, unorthodox insights to his writing that will stimulate readers (from high schoolers to emeritus professors) to think about old topics in new, nonstale ways."---SIAM Review --
"Stillwell weaves historical details into his writing seamlessly, helping to give the reader the true feeling that mathematics is more than just a bunch of people playing games with symbols, but rather a rich and rewarding intellectual endeavor important to the human enterprise."---MAA Reviews --
While many popular books have been written on the advances in our understanding of infinity, sparked by the set theory of Georg Cantor in the 1870s and incompleteness theorems of Kurt Godel in the 1930s, such books generally dwell on a single aspect of either set theory or logic. The aim of this book is to explain the whole, in which set theory interacts with logic, and both begin to affect mainstream mathematics (the latter being quite a recent development, not yet given much space in popular accounts). --
In Roads to Infinity, award-winning author John Stillwell explores the consequences of accepting infinity, which are rich and surprising. The reader needs very little background beyond high school mathematics, but should have a willingness to grapple with alien ideas. Stillwell's style eases the reader into the technicalities of set theory and logic by tracing a single thread in each chapter, beginning with a natural mathematical question and following a sequence of historic responses to that question. Each response typically leads to new questions, and from them new concepts and theorems emerge. At the end of each chapter a section called "Historical Background" situates the thread in a bigger picture of mathematics and its history. --
By following this path, key ideas are presented first, then revisited and reinforced by showing a wider view. Some readers, however, may be impatient to get to the core theorems and will skip the historical background sections, at least at first reading. Others, in search of a big picture from the beginning, may begin by reading the historical background and then come back to fill in details. --Book Jacket
LC subject Set theory
Logic, Symbolic and mathematical
ISBN 9781568814667 (alk. paper)
1568814666 (alk. paper)