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 entscheidungsproblem ; 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 PA ; Embedding PA and PA; Cut elimination in PA ; 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.
