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Understanding the Infinite

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In writing this book I have tried to keep mathematical prerequisites to a minimum. The reader who is essentially innocent of mathematical knowledge beyond that taught in high school should be able to read at least halfway through Chapter VIII plus parts of the rest of the book, though such a reader will need to skip the occasional formula. That is enough of the book for all of
the major ideas to be presented. The Introduction may seem daunting since t refers to ideas that are not explained until later-trust me, they are explained. A reader who learned freshman calculus once, but perhaps does not remember it very well, and who has had a logic course that included a proof of the completeness theorem will be in fine shape throughout the book, except for various "technical remarks," an appendix to Chapter VI, and a few parts of Chapter IX. Those few technical discussions require varying degrees of mathematical sophistication and knowledge of general mathematical logic plus occasional knowledge of elementary recursion theory, model theory, or modal logic.

Thanks are due Bonnie Kent, Vann McGee, Sidney Morgenbesser, and Sarah Stebbins for their infinite patience in listening to my many half-baked ideas and for their substantial help in culling and completing them while I was writing this book. As they learned, I cannot think without the give and take of conversation. Thanks also to Ti-Grace Atkinson, Jeff Barrett, William Boos, Hartry Field, Alan Gabbey, Haim Gaifman, Alexander George, Allen Hazen, Gregory Landini, Penelope Maddy, Robert Miller, Edward Nelson, Ahmet Omurtag, David Owen, Charles Parsons, Thomas Pogge, Vincent Renzi, Scott Shapiro, Mark Steiner, and Robert Vaught for their thoughtful comments on an early version of the book. Those comments have led to significant improvements. And thanks to Thomas Pogge for his substantial helpin correcting my translations from German. Any remaining mistakes are, of course, my own.

Thanks are also due my parents, Dorothy and Leroy Lavine, not only for their moral support, which I very much appreciated, but also for their generous financial support, without which the preparation of this book would not have been possible. My wife, Caroline, and daughter, Caila, deserve the most special thanks of all, for tolerating with such understanding my absences and the stresses on our family life that the writing of a book inevitably required. This book is dedicated to them.
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