Roads to Infinity: The Mathematics of Truth and Proof (Hardcover)

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
Godel’s First Incompleteness Theorem
Godel’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