Incompleteness in the Land of Sets

Incompleteness in the Land of Sets
Author: Melvin Fitting
Publsiher: Unknown
Total Pages: 0
Release: 2007
Genre: Incompleteness theorems
ISBN: 1904987346

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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.

Forever Undecided

Forever Undecided
Author: Raymond M. Smullyan
Publsiher: Knopf
Total Pages: 257
Release: 2012-07-04
Genre: Mathematics
ISBN: 9780307962461

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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!

G del s Theorem

G  del s Theorem
Author: Torkel Franzén
Publsiher: CRC Press
Total Pages: 182
Release: 2005-06-06
Genre: Mathematics
ISBN: 9781439876923

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"Among the many expositions of Gödel's incompleteness theorems written for non-specialists, this book stands apart. With exceptional clarity, Franzén gives careful, non-technical explanations both of what those theorems say and, more importantly, what they do not. No other book aims, as his does, to address in detail the misunderstandings and abuses of the incompleteness theorems that are so rife in popular discussions of their significance. As an antidote to the many spurious appeals to incompleteness in theological, anti-mechanist and post-modernist debates, it is a valuable addition to the literature." --- John W. Dawson, author of Logical Dilemmas: The Life and Work of Kurt Gödel

The Theory of G del

The Theory of G  del
Author: Carlo Cellucci
Publsiher: Springer Nature
Total Pages: 164
Release: 2022-11-21
Genre: Philosophy
ISBN: 9783031134173

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This book presents Gödel’s incompleteness theorems and the other limitative results which are most significant for the philosophy of mathematics. Results are stated in the form most relevant for use in the philosophy of mathematics. An appendix considers their implications for Hilbert’s Program for the foundations of mathematics. The text is self-contained, all notions being explained in full detail, but of course previous exposure to the very first rudiments of mathematical logic will help.

Raymond Smullyan on Self Reference

Raymond Smullyan on Self Reference
Author: Melvin Fitting,Brian Rayman
Publsiher: Springer
Total Pages: 195
Release: 2018-01-11
Genre: Philosophy
ISBN: 9783319687322

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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.

Universality in Set Theories

Universality in Set Theories
Author: Manuel Bremer
Publsiher: Walter de Gruyter
Total Pages: 125
Release: 2013-05-02
Genre: Philosophy
ISBN: 9783110326109

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The book discusses the fate of universality and a universal set in several set theories. The book aims at a philosophical study of ontological and conceptual questions around set theory. Set theories are ontologies. They posit sets and claim that these exhibit the essential properties laid down in the set theoretical axioms. Collecting these postulated entities quantified over poses the problem of universality. Is the collection of the set theoretical entities itself a set theoretical entity? What does it mean if it is, and what does it mean if it is not? To answer these questions involves developing a theory of the universal set. We have to ask: Are there different aspects to universality in set theory, which stand in conflict to each other? May inconsistency be the price to pay to circumvent ineffability? And most importantly: How far can axiomatic ontology take us out of the problems around universality?

Canada s Odyssey

Canada s Odyssey
Author: Peter H. Russell
Publsiher: University of Toronto Press
Total Pages: 544
Release: 2017-05-08
Genre: History
ISBN: 9781487514488

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150 years after Confederation, Canada is known around the world for its social diversity and its commitment to principles of multiculturalism. But the road to contemporary Canada is a winding one, a story of division and conflict as well as union and accommodation. In Canada’s Odyssey, renowned scholar Peter H. Russell provides an expansive, accessible account of Canadian history from the pre-Confederation period to the present day. By focusing on what he calls the "three pillars" of English Canada, French Canada, and Aboriginal Canada, Russell advances an important view of our country as one founded on and informed by "incomplete conquests". It is the very incompleteness of these conquests that have made Canada what it is today, not just a multicultural society but a multinational one. Featuring the scope and vivid characterizations of an epic novel, Canada’s Odyssey is a magisterial work by an astute observer of Canadian politics and history, a perfect book to commemorate the 150th anniversary of Confederation.

Fundamentals of Mathematical Logic

Fundamentals of Mathematical Logic
Author: Peter G. Hinman
Publsiher: CRC Press
Total Pages: 894
Release: 2018-10-08
Genre: Mathematics
ISBN: 9781439864272

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This introductory graduate text covers modern mathematical logic from propositional, first-order and infinitary logic and Gödel's Incompleteness Theorems to extensive introductions to set theory, model theory and recursion (computability) theory. Based on the author's more than 35 years of teaching experience, the book develops students' intuition by presenting complex ideas in the simplest context for which they make sense. The book is appropriate for use as a classroom text, for self-study, and as a reference on the state of modern logic.