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Incompleteness in the Land of Sets

Author: Melvin Fitting

Publisher:

ISBN:

Category: Mathematics

Page: 142

View: 210

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 Godel, 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. Godel 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 Godel's Second Incompleteness Theorem. Exercises are scattered throughout.

Raymond Smullyan on Self Reference

Author: Melvin Fitting

Publisher: Springer

ISBN:

Category: Philosophy

Page: 195

View: 686

This book collects, for the first time in one volume, contributions honoring Professor Raymond Smullyan’s work on self-reference. It serves not only as a tribute to one of the great thinkers in logic, but also as a celebration of self-reference in general, to be enjoyed by all lovers of this field. Raymond Smullyan, mathematician, philosopher, musician and inventor of logic puzzles, made a lasting impact on the study of mathematical logic; accordingly, this book spans the many personalities through which Professor Smullyan operated, offering extensions and re-evaluations of his academic work on self-reference, applying self-referential logic to art and nature, and lastly, offering new puzzles designed to communicate otherwise esoteric concepts in mathematical logic, in the manner for which Professor Smullyan was so well known. This book is suitable for students, scholars and logicians who are interested in learning more about Raymond Smullyan's work and life.

The Bulletin of Symbolic Logic

Author:

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Category: Electronic journals

Page:

View: 364

A Discourse and Register Analysis of the Prophetic Book of Joel

Author: Colin Toffelmire

Publisher: BRILL

ISBN:

Category: Religion

Page: 242

View: 764

In A Discourse and Register Analysis of the Prophetic Book of Joel, Colin M. Toffelmire presents a thorough analysis of the text of Joel from the perspective of Systemic Functional Linguistics.

Godel's Incompleteness Theorems

Author: Raymond M. Smullyan

Publisher: Oxford University Press

ISBN:

Category: Mathematics

Page: 160

View: 535

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.

Social Mobility in Traditional China

Author:

Publisher: Brill Archive

ISBN:

Category:

Page:

View: 242

Sets, Logic and Categories

Author: Peter J. Cameron

Publisher: Springer Science & Business Media

ISBN:

Category: Mathematics

Page: 182

View: 961

Set theory, logic and category theory lie at the foundations of mathematics, and have a dramatic effect on the mathematics that we do, through the Axiom of Choice, Gödel's Theorem, and the Skolem Paradox. But they are also rich mathematical theories in their own right, contributing techniques and results to working mathematicians such as the Compactness Theorem and module categories. The book is aimed at those who know some mathematics and want to know more about its building blocks. Set theory is first treated naively an axiomatic treatment is given after the basics of first-order logic have been introduced. The discussion is su pported by a wide range of exercises. The final chapter touches on philosophical issues. The book is supported by a World Wibe Web site containing a variety of supplementary material.

The Law Magazine and Review

Or Quarterty Journal of Jurisprudence

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ISBN:

Category:

Page:

View: 820

Dream-land by Daylight

A Panorama of Romance

Author: Caroline Chesebro'

Publisher:

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Category: Short stories, American

Page: 428

View: 536

The Queensland Law Society Journal

Author:

Publisher:

ISBN:

Category: Bar associations

Page:

View: 570

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