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Roads to Infinity: The Mathematics of Truth and Proof

John Stillwell
Publisher: 
A K Peters
Publication Date: 
2010
Number of Pages: 
203
Format: 
Hardcover
Price: 
39.00
ISBN: 
9781568814667
Category: 
General
BLL Rating: 

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Allen Stenger
, on
09/14/2010
]

This book is an interesting sampler from the foundations of mathematics. It has the flavor of a series of colloquium lectures: some things are examined in detail, there are some proofs, and the rest of it is background material to give you the context and to sketch the boundaries of the subject. The formal prerequisites are low (high-school math), but many parts of the book are quite difficult with some intricate reasoning. You can browse and read selectively for the parts that interest you; the lengthy “Historical Background” sections are especially good.

Despite the title, the book is less about infinity that it is about logic, especially completeness and incompleteness theorems and proofs of consistency. The book attempts to weave together logic and infinity, but I thought this was only partly successful; most of the discussion is either 100% infinity or 100% logic. The most interesting part of the book for me was the chapter on natural unprovable statements.

The book is an impressive accomplishment: It is well-written, it packs a comprehensive overview and some serious and detailed mathematics into 200 pages, and it is structured so that it can be read at several different levels. The most serious limitation is that it sticks strictly to the mainstream approach to these subjects. There’s no mention of how controversial Cantor’s theory of infinity was when it first came out, or of the vicious attacks on it by Kronecker and others. There’s also almost no mention of intuitionism and constructivism, which have their own very different views of infinity and logic. Including more discussion of these areas would have helped illuminate why the subjects have developed as they have.


Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.

 


  • PREFACE
  1. THE DIAGONAL ARGUMENT
    1. Counting and Countability
    2. Does One Infinite Size Fit All?
    3. Cantor’s Diagonal Argument
    4. Transcendental Numbers
    5. Other Uncountability Proofs
    6. Rates of Growth
    7. The Cardinality of the Continuum
    8. Historical Background
  2. ORDINALS
    1. Counting Past Infinity
    2. The Countable Ordinals
    3. The Axiom of Choice
    4. The Continuum Hypothesis
    5. Induction
    6. Cantor Normal Form
    7. Goodstein’s Theorem
    8. Hercules and the Hydra
    9. Historical Background
  3. COMPUTABILITY AND PROOF
    1. Formal Systems
    2. Post’s Approach to Incompleteness
    3. Gödel’s First Incompleteness Theorem.
    4. Gödel’s Second Incompleteness Theorem
    5. Formalization of Computability
    6. The Halting Problem
    7. The Entscheidungsproblem
    8. Historical Background
  4. LOGIC
    1. Propositional Logic
    2. A Classical System
    3. A Cut-Free System for Propositional Logic
    4. Happy Endings
    5. Predicate logic
    6. Completeness, Consistency, Happy Endings
    7. Historical Background
  5. ARITHMETIC
    1. How Might We Prove Consistency?
    2. Formal Arithmetic
    3. The Systems PA and PAω
    4. Embedding PA in PAω
    5. Cut Elimination in PAω
    6. The Height of This Great Argument
    7. Roads to Infinity
    8. Historical Background
  6. NATURAL UNPROVABLE SENTENCES
    1. A Generalized Goodstein Theorem
    2. Countable Ordinals via Natural Numbers
    3. From Generalized Goodstein to Well-Ordering
    4. Generalized and Ordinary Goodstein
    5. Provably Computable Functions
    6. Complete Disorder Is Impossible
    7. The Hardest Theorem in Graph Theory
    8. Historical Background
  7. AXIOMS OF INFINITY
    1. Set Theory without Infinity
    2. Inaccessible Cardinals
    3. The Axiom of Determinacy
    4. Largeness Axioms for Arithmetic
    5. Large Cardinals and Finite Mathematics
    6. Historical Background
  • BIBLIOGRAPHY
  • INDEX

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