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Roads to Infinity

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Table of Contents

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 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 Computability and Proof Formal Systems Post's Approach to Incompleteness Goedel's First Incompleteness Theorem Goedel's Second Incompleteness Theorem Formalization of Computability The Halting Problem The Entscheidungs problem Historical Background Logic Propositional Logic A Classical System A Cut-Free System for Propositional Logic Happy Endings Predicate Logic Completeness, Consistency, Happy Endings Historical Background Arithmetic How Might We Prove Consistency? Formal Arithmetic The Systems PA and PAÏ Embedding PA in PAÏ Cut Elimination in PAÏ The Height of This Great Argument Roads to Infinity Historical Background Natural Unprovable Sentences A Generalized Goodstein Theorem Countable Ordinals via Natural Numbers From Generalized Goodstein to Well-Ordering Generalized and Ordinary Goodstein Provably Computable Functions Complete Disorder Is Impossible The Hardest Theorem in Graph Theory Historical Background Axioms of Infinity Set Theory without Infinity Inaccessible Cardinals The Axiom of Determinacy Largeness Axioms for Arithmetic Large Cardinals and Finite Mathematics Historical Background

About the Author

John Stillwell was born in Melbourne, Australia in 1942 and educated at Melbourne High School, the University of Melbourne (M.Sc. 1965), and MIT (Ph.D. 1970). From 1970 to 2001 he taught at Monash University in Melbourne, and since 2002 he has been Professor of Mathematics at the University of San Francisco. He has been an invited speaker at several international conferences, including the International Congress of Mathematicians in Zurich 1994. His works cover a wide spectrum of mathematics, from translations of classics by Dirichlet, Dedekind, Poincare, and Dehn to books on algebra, geometry, topology, number theory, and their history. For his expository writing he was awarded the Chauvenet Prize of the Mathematical Association of America in 2005, and the AJCU National Book Award in 2009. Recent titles by Stillwell include Yearning for the Impossible, Mathematics and Its History, The Four Pillars of Geometry, and Geometry of Surfaces.


I highly recommend it for undergraduates in mathematics and other young mathematicians who are looking for historical context or a different angle to their studies. Readers who have experience with theoretical analysis or a foundation in abstract mathematics will find the examples wonderfully illustrative. For these readers, Stillwell's words will flow smoothly, almost like a novel. --Joyance Meechai, Mathematics Teacher, October 2011 In 1963, Edwin E. Moise published Elementary Geometry from an Advanced Standpoint and his book became a classic. ! [this book] deserves the same outcome. ! One of the most enjoyable features is Stillwell's use of techniques of logic and set theory to solve real mathematical problems ! Another enjoyable feature is Stillwell's uniform coverage of unprovability, undecidability and non-computability ! suitable for self-study ! it is excellent background material for computer scientists and mathematicians in other fields. The historical notes alone are worth perusing by anyone who is interested in the development of mathematical ideas. --Phill Schultz, Gazette of the Australian Mathematical Society, March 2011 ! a clear and succinct guide. ! One interesting feature of the book is the careful treatment of two of the less famous contributors in this area--Emil Post and Gerhard Gentzen ! --CMS Notes, Vol. 43, No. 1, February 2011 ! excellent book ! the investment the reader makes--be he an intellectually curious adult or a math grad student with extra time on her hands--pays off with an increased understanding of the fascinating world of mathematical logic. The author's thorough, well-researched historical comments are particularly valuable, as well as the philosophical quotations from the important players in this game. There is a very complete bibliography. What the reader might appreciate most is the ability of the author to share his deep insights into what is important and what it all means in the most profound sense. ! it is clear that the book received excellent proofreading before publication. ! --Mathematical Reviews, Issue 2011f Featuring chapters dedicated to the diagonal argument, ordinals, computability and proof, logic, arithmetic, natural unprovable sentences, and axioms, as well as being enhanced with the inclusion of a lengthy bibliography and a comprehensive index, Roads to Infinity: The Mathematics of Truth and Proof is highly recommended reading for students, scholars, and non-specialist general readers with an interest in the history and contemporary issues of mathematics today. --Able Greenspan, The Midwest Book Review I love reading anything by John Stillwell. If you've ever been tantalized by the puzzles of infinity, set theory, and logic, and want to understand what's really going on, this is the book for you. It's an exceptionally fine piece of mathematical exposition. --Steven Strogatz, Cornell University, author of The Calculus of Friendship

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