
Incompleteness in the Land of Sets
M. Fitting(Author)
College Publications (Publisher)
Published on 19. February 2007
Book
Paperback/Softback
156 pages
978-1-904987-34-5 (ISBN)
Description
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 Gödel, 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. Gödel 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 Gödel's Second Incompleteness Theorem. Exercises are scattered throughout.
More details
Language
English
Place of publication
London
United Kingdom
Target group
Adult education
Professional and scholarly
Product notice
Paperback (trade)
Illustrations
black & white illustrations
Dimensions
Height: 234 mm
Width: 156 mm
Thickness: 9 mm
Weight
248 gr
ISBN-13
978-1-904987-34-5 (9781904987345)
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Schweitzer Classification