Roads to Infinity: The Mathematics of Truth and Proof 1st Edition by John C. Stillwell (PDF)

Description

Winner of a CHOICE Outstanding Academic Title Award for 2011!This book offers an introduction to modern ideas about infinity and their implications for mathematics. It unifies ideas from set theory and mathematical logic, and traces their effects on mainstream mathematical topics of today, such as number theory and combinatorics. The treatment is historical and partly informal, but with due attention to the subtleties of the subject. Ideas are shown to evolve from natural mathematical questions about the nature of infinity and the nature of proof, set against a background of broader questions and developments in mathematics. A particular aim of the book is to acknowledge some important but neglected figures in the history of infinity, such as Post and Gentzen, alongside the recognized giants Cantor and Gödel.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐John Stillwell’s “Roads to Infinity” sounds like another “Gödel’s theorem” book. Yes, it mentions it and gets into Gödel’s famous theorems, a little bit. But, he just finds a simple alternative; he doesn’t mention Gödel numbering. I’ve been through say Nagel’s “Gödel’s Proof.” But, even then, I don’t recall all this stuff about ordinals. John Stillwell shows George Cantor’s genius a lot more by showing the Fourier analyses roots of his ideas and George Cantor’s ordinal numbers work. This is just a start of what you can get from here, if you’re not a phd logician/mathematician.If you think George Cantor’s transfinite numbers is mindblowing, his work on ordinals to analyse infinit cardinal numbers is . . . it shows how one can analyse infinity far more than most people could ever dream.I’d like to say, that as usual, in John Stillwell’s works, he always finds some modern easier short proof of results; otherwise, he references off. He gives examples when he can find easy examples, and just mentions hard problems(mostly in chapter supplements where they belong).In John Stillwell’s other works, he loves to point out the induction definitions of arithmetic; in this book, he gives a lot more reason to take those seriously!Historically, one of the major reasons for proof was the inability to deal with infinity. Here, John Stillwell shows the affect infinity has had on proof. I think logical proof has proven to be more than we thought through the ages, so I don’t think one should consider syllogisms discredited by modern studies of infinity and proof. But, I do like the recent results of the interplay between finite and infinit. And John’s book . . . well, I don’t think he does more than introduce one to these things.In Oyestein Ore’s “Number theory and its History”, he gets into some number theory, some diophanine analyses, where he shows some lead to infinity and some are finite. This has always suggested to me that determining when things are finite and when they are infinity, when they are compact or not seems to be an interesting question in mathematics(I can’t say that I totally know lie theory; but, I have done some look ahead and found that some of the majory lie theory work of the twentieth century anyways has to do with compact and non-compact geometry). And so, the mathematics of the interplay between the finit and infinit at least excited me.

⭐This book is very special in tying together the concepts of infinity, logic and computation, with a lot of clarity. I believe that a prior encounter with each of those topics would be a prerequisite for many or most readers. The book is both rigorous and providing the historical and psychological/cognitive aspects of how things came to be, allowing the philosophical gaps inherent to the standardized theory to become apparent to the avid reader.As a result, and in general, its treatment of incompleteness is one of the best I’ve come across to date.Some specific points could have been made more explicit/detailed in the opening chapters of the book, such as going into detail into variants of the initial diagonalization argument, which are the core to much of the rest to come, as they are probably hard to come across elsewhere perhaps except for maybe in other books by the same author or in dusty arcane books on the subject. Hence a 4 stars.I am looking forward to completing reading the book, and then going through a second read of it!

⭐I bought this book out of interest since it had very good reviews from other readers. I was not disappointing as the book is very well written, informative, and interesting. The introductory chapters on the concept of infinity provide very clear explanations of the problems associated with defining what is the limit of very large numbers. This book I would highly recommend to anybody who wanted to read a non-technical account of the issues around infinity and to come away with a sound understanding of the what the term means.

⭐Whoa… I mean this is the kind of book that is so significant and intellect changing that you simply MUST read it, but…….and this is a big but, read slowly, take your time, and absorb. Utterly mind stretching……….I LOVED IT! Will be reading this many times……

⭐This is an excellent introduction to how logic and modern set theory affect modern mathematics. Delves into Cantor’s theory of transfinite numbers. Also examines work of Kurt Godel, Emil Post , and Gerhard Gentzen. Not too technical. Very readable.

⭐Clear and very concise. Stillwell did a masterful job on this review of the mathematics of different infinities.The book is both content rich and also easy to understand.

⭐The book’s writing sometimes seemed vague and confusing. That said, it did have interesting content.

⭐I love Stillwell’s books and would be great to collect but the prices of these books are close to usury. CRC books are nothing short of excellent but man these prices!!!!!

⭐This is a prequal to Stillwells book on analysis and set tbeory. It places into context a major problem in mathematics that we have today, that of a working tbeory that addresses our failed attempts to model the continuum. Stillwell tackles the problem head on in tnis book and in his sequel, both highly recommended. Modern books on analysis fail completely to explain how the real numbers are constructed and the problems surrounding the irrational numbers. I suspect it is because that the powers that be in tne mathematical world are reluctant to admit tbat they do not have a solution and blindly accept the ideas of axiomatuc set theory in providing a workable arithmetic of real numbers. This book gets to the heart of tbe problem and adresses the issues faced by the ancient Greek mathematicians to the mathemeticians of today. It provides no solutions but does uncover tbe weakness in the arguments put forward and that are still taught in Universities across the globe when dealing with countable and uncountable sets of ordinals. This book along with its sequel is a must for the archives as I am sure in years to come that it will be viewed as the first text that enlightens the mathematics world to where it has taken a wrong path.

⭐Good book, great condition as described.

⭐Price is a bit high, hence my rating.Didactically-talented, Stillwell gives a very good exposition of ordinals, along with Goodstein’s theorem ; with emphasis on Emil Post’s and Gerhard Gentzen’s discoveries and proofs, with their relations to Gödel’s theorems…All in all, an excellent overwiew as well as inroads to recent discoveries based on large cardinals…

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