Roads to infinity: the mathematics of truth and proof

Abstract

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 PAomega Embedding PA in PAomega Cut Elimination in PAomega 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