Godel's Incompleteness Theorems
Kurt Godel, the greatest logician of our time, startled the world of mathematics in 1931 with his Theorem of Undecidability, which showed that some statements in mathematics are inherently "undecidable." His work on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum theory brought him further worldwide fame. In this introductory volume, Raymond Smullyan, himself a well-known logician, guides the reader through the fascinating world of Godel's incompleteness theorems. The level of presentation is suitable for anyone with a basic acquaintance with mathematical logic. As a clear, concise introduction to a difficult but essential subject, the book will appeal to mathematicians, philosophers, and computer scientists.
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Arithmetic set Assume hypothesis atomic formula axiom scheme axiom system B(BX base 10 notation base 13 believe Bp BX is provable called conditions hold consistent axiomatizable Corollary disjoint Eo-formula Exercise expresses the set first-order logic formula F formula F(vi free occurrences free variable Godel number Godel sentence Godel's incompleteness theorem H(vi Hence incompleteness proof incompleteness theorems inconsistent induction Knights and Knaves last chapter Lemma mathematical induction modal logic modus ponens native natural numbers never believe number n ordered pairs Peano Arithmetic printable proof of Theorem propositional logic provability predicate provable in P.A. provable sentences reasoner believes reasoner of type recursive refutable representable Rosser system sentence F(n sentence G sequence number set of Godel Si-sets simply consistent So-sentences are provable strongly definable subsystem Suppose symbols system P.A. Tarski's theorem true iff true sentence true So-sentences truth undecidable w-consistent X D Y xn,y