Forever undecided : a puzzle guide to Gödel

A challenging puzzle collection and an instructive and entertaining introduction to Kurt Godel's famous theorems, including incompleteness and undecidability. Much of the action of the book takes place on an imaginary and magical island, the Island of Knights and Knaves, where knights always make true statements, knaves always make false statements, and every inhabitant is either a knight or a knave. Here we meet an amazing array of characters, visitors to the island, seeking to determine the natives' true identities. Among them are the census-taker McGregor; a philosopher-logician in search of his flighty bird-wife Oona; and a regiment of Reasoners. By following the Reasoners through brain-tingling exercises and adventures - including journeys into the "other possible worlds" of Kripke semantics - even the most illogical of us should come to understand Godel's theorems, some of their philosophical and mathematical implications and why we, like Godel himself, must remain forever undecided! The book is intended for puzzle fans of every age and ability - from the high-school whizz to the seasoned mathematician, logician or computer scientist. Forever Undecided is the most challenging yet of Raymond Smullyan's puzzle collections. It is, at the same time, an introduction—ingenious, instructive, entertaining—to Gödel's famous theorems. With all the wit and charm that have delighted readers of his previous books, Smullyan transports us once again to that magical island where knights always tell the truth and knaves always lie. Here we meet a new and amazing array of characters, visitors to the island, seeking to determine the natives'identities. Among them: the census-taker McGregor; a philosophical-logician in search of his flighty bird-wife, Oona; and a regiment of Reasoners (timid ones, normal ones, conceited, modest, and peculiar ones) armed with the rules of propositional logic (if X is true, then so is Y). By following the Reasoners through brain-tingling exercises and adventures—including journeys into the “other possible worlds” of Kripke semantics—even the most illogical of us come to understand Gödel's two great theorems on incompleteness and undecidability, some of their philosophical and mathematical implications, and why we, like Gödel himself, must remain Forever Undecided! Contents......Page 9 Preface......Page 13 Part I : YOU MIGHT BE SURPRISED!......Page 15 1 A DiabolicalPuzzle......Page 17 2 Surprised?......Page 22 Part II : THE LOGIC OF LYING AND TRUTH TELLING......Page 27 3 The CensusTaker......Page 29 4 In Search ofOona......Page 37 5 An InterplanetaryTangle......Page 42 Part III : KNIGHTS, KNAVES,AND PROPOSITIONALLOGIC......Page 49 6 A Bit ofPropositionalLogic......Page 51 7 Knights, Knaves,and PropositionalLogic......Page 62 8 Logical Closureand Consistency......Page 69 Part IV: LET'S BECAREFUL!......Page 79 9 Paradoxical?......Page 81 10 The ProblemDeepens......Page 91 Part V:THE CONSISTENCYPREDICAMENT......Page 101 11 LogiciansWho Reason AboutThemselves......Page 103 12 The ConsistencyPredicament......Page 114 13 GodelianSystems......Page 121 14 More ConsistencyPredicatnents......Page 126 Part VI : SELF -FULFILLINGBELIEFS ANDLOB'S THEOREM......Page 135 15 Self-FulfillingBeliefs......Page 137 A collection of puzzles which provides an introduction to Godel's theorems, transferring them from the formal domain of mathematical systems in an attempt to make the essentials of his ideas more accessible to the general reader.