Incompleteness in the Land of Sets (Paperback) (ISBN-13: 9781904987345)

Russell's paradox arises when we consider those sets that do not belong to themselves. The collection of such sets cannot constitute a set. Step back a bit. Logical formulas define sets (in a standard model). Formulas, being mathematical objects, can be thought of as sets themselves-mathematics reduces to set theory. Consider those formulas that do not belong to the set they define. The collection of such formulas is not definable by a formula, by the same argument that Russell used. This quickly gives Tarski's result on the undefinability of truth. Variations on the same idea yield the famous results of G del, Church, Rosser, and Post. This book gives a full presentation of the basic incompleteness and undecidability theorems of mathematical logic in the framework of set theory. Corresponding results for arithmetic follow easily, and are also given. G del numbering is generally avoided, except when an explicit connection is made between set theory and arithmetic. The book assumes little technical background from the reader. One needs mathematical ability, a general familiarity with formal logic, and an understanding of the completeness theorem, though not its proof. All else is developed and formally proved, from Tarski's Theorem to G del's Second Incompleteness Theorem. Exercises are scattered throughout.

Author: M. Fitting
ISBN-10: 1904987346
ISBN-13: 9781904987345
Publisher: College Publications
Language: English
Published: 02/19/2007
Pages: 156
Format: Paperback
Weight: 0.50lbs
Size: 9.21h x 6.14w x 0.33d