Roads to Infinity: The Mathematics of Truth and Proof

Roads to Infinity: The Mathematics of Truth and Proof

by John C. Stillwell
     
 

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ISBN-10: 1568814666

ISBN-13: 9781568814667

Pub. Date: 07/13/2010

Publisher: Taylor & Francis

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

Overview

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.

Product Details

ISBN-13:
9781568814667
Publisher:
Taylor & Francis
Publication date:
07/13/2010
Pages:
250
Sales rank:
900,728
Product dimensions:
6.00(w) x 9.10(h) x 0.90(d)

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
Gödel’s First Incompleteness Theorem
Gödel’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 PAω
Embedding PA in PAω
Cut Elimination in PAω
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

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