Forever UndecidedKnopf Doubleday Publishing Group, 04.07.2012 - 257 Seiten 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! |
Inhalt
In Search of Oona | |
Paradoxical? | |
The Problem Deepens | |
Logicians Who Reason About Themselves | |
The Consistency Predicament | |
SelfFulfilling Beliefs | |
The Rajahs Diamond | |
Löbs Island | |
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ancestor assumption axioms of K4 B(pq become inconsistent believe Bp believe k believe q believes the professor believes the proposition believes the rules Corollary correctly believe cure diamond established propositions Exercise fact false proposition formula George Boolos Gödel number Gödelian system Island of Knights knight-knave island Knights and Knaves Kripke model Lemma Löb's Theorem logical consequence logically equivalent logician Martian mathematical systems means minimal reasoner modal logic modal system modest reasoner modest with respect modus ponens native says necessarily true never believe Oona peculiar possible worlds pƆq printable printed Prize proof proposition q propositional logic provable in G prove reasoner believes reasoner of type reasoner will believe reflexive reasoner self-referentially correct sentences provable statement is false student Suppose a native Suppose a reasoner symbol system G system of type tautology true for M2 true proposition true statement truth table type 4 believes type G Venusian