In 1931 kurt gödel published his fundamental paper, "On Formally Undecidable Propositions of Principia Mathematica and Related Systems. This revolutionary paper challenged certain basic assumptions underlying much research in mathematics and logic. With a new introduction by Douglas R. Ernest nagel and james newman provide a readable and accessible explanation to both scholars and non-specialists of the main ideas and broad implications of Gödel's discovery.

Godel's Proof #ad - It offers every educated person with a taste for logic and philosophy the chance to understand a previously difficult and inaccessible subject. Marking the 50th anniversary of the original publication of Gödel's Proof, New York University Press is proud to publish this special anniversary edition of one of its bestselling and most frequently translated books.

The award committee described his work in mathematical logic as "one of the greatest contributions to the sciences in recent times. However, few mathematicians of the time were equipped to understand the young scholar's complex proof.

# On Formally Undecidable Propositions of Principia Mathematica and Related Systems Dover Books on Mathematics

Dover Publications #ad - It is thus uncertain that the basic axioms of arithmetic will not give rise to contradictions. Not only does it make the argument more intelligible, but the introduction contributed by Professor R. Meltzer, University of Edinburgh. Braithwaite. Introduction by R. The repercussions of this discovery are still being felt and debated in 20th-century mathematics.The present volume reprints the first English translation of Giidel's far-reaching work. Translated by B. B. Preface. Braithwaite cambridge university}, an excellent work of scholarship in its own right, illuminates it by paraphrasing the major part of the argument. This dover edition thus makes widely available a superb edition of a classic work of original thought, one that will be of profound interest to mathematicians, logicians and anyone interested in the history of attempts to establish axioms that would provide a rigorous basis for all mathematics.

On Formally Undecidable Propositions of Principia Mathematica and Related Systems Dover Books on Mathematics #ad - B. Kurt giidel maintained, that in any arithmetic system, and offered detailed proof, even in elementary parts of arithmetic, there are propositions which cannot be proved or disproved within the system. In 1931, a young austrian mathematician published an epoch-making paper containing one of the most revolutionary ideas in logic since Aristotle.

# Incompleteness: The Proof and Paradox of Kurt Gödel Great Discoveries: The Proof and Paradox of Kurt Godel

W. W. Norton & Company #ad - A gem…an unforgettable account of one of the great moments in the history of human thought. Steven pinkerprobing the life and work of Kurt Gödel, Incompleteness indelibly portrays the tortured genius whose vision rocked the stability of mathematical reasoning—and brought him to the edge of madness.# I Am a Strange Loop

Basic Books #ad - The most central and complex symbol in your brain is the one called "I. The "i" is the nexus in our brain, one of many symbols seeming to have free will and to have gained the paradoxical ability to push particles around, rather than the reverse. One of our greatest philosophers and scientists of the mind asks, where does the self come from--and how our selves can exist in the minds of others.Compulsively readable and endlessly thought-provoking, this is a moving and profound inquiry into the nature of mind. How can a mysterious abstraction be real-or is our "i" merely a convenient fiction? does an "I" exert genuine power over the particles in our brain, or is it helplessly pushed around by the laws of physics? These are the mysteries tackled in I Am a Strange Loop, Douglas Hofstadter's first book-length journey into philosophy since Gödel, Escher, Bach.

I Am a Strange Loop #ad - Can thought arise out of matter? can self, "i" arise out of mere matter? if it cannot, consciousness, soul, then how can you or I be here? I Am a Strange Loop argues that the key to understanding selves and consciousness is the "strange loop"-a special kind of abstract feedback loop inhabiting our brains.

# A Profile of Mathematical Logic Dover Books on Mathematics

Dover Publications #ad - A treat for both the intellect and the imagination, it profiles the development of logic from ancient to modern times and compellingly examines the nature of logic and its philosophical implications. Anyone seeking a readable and relatively brief guide to logic can do no better than this classic introduction.. No prior knowledge of logic is necessary; readers need only an acquaintance with high school mathematics. The treatment of the gödel metatheorems is especially detailed and clear, and answers to the problems appear at the end. The author emphasizes understanding, the rise of mathematical logic after more than 2, the nature of the formal axiomatic method and the reasons for its use, 000 years of traditional logic, rather than technique, and focuses on such topics as the historical reasons for the formation of Aristotelian logic, and the main results of metatheory and their philosophic import.

# A World Without Time: The Forgotten Legacy of Godel and Einstein

Basic Books #ad - By 1949, godel had produced a remarkable proof: In any universe described by the Theory of Relativity, time cannot exist. Yet cosmologists and philosophers alike have proceeded as if this discovery was never made. In 1942, exchanging ideas about science, politics, philosophy, the logician Kurt Godel and Albert Einstein became close friends; they walked to and from their offices every day, and the lost world of German science.. Einstein endorsed this result reluctantly but he could find no way to refute it, since then, neither has anyone else. In a world without time, telling the story of two magnificent minds put on the shelf by the scientific fashions of their day, Palle Yourgrau sets out to restore Godel to his rightful place in history, and attempts to rescue the brilliant work they did together.

# The Gödelian Puzzle Book: Puzzles, Paradoxes and Proofs

Dover Publications #ad - The first three chapters of Part II contain generalized Gödel theorems. These brand-new recreational logic puzzles provide entertaining variations on Gödel's incompleteness theorems, offering ingenious challenges related to infinity, undecidability, truth and provability, and other concepts. Created by the celebrated logician Raymond Smullyan, the puzzles require no background in formal logic and will delight readers of all ages.The two-part selection of puzzles and paradoxes begins with examinations of the nature of infinity and some curious systems related to Gödel's theorem. Symbolic logic is deferred until the last three chapters, which give explanations and examples of first-order arithmetic, Peano arithmetic, and a complete proof of Gödel's celebrated result involving statements that cannot be proved or disproved.

The Gödelian Puzzle Book: Puzzles, Paradoxes and Proofs #ad - The book also includes a lively look at decision theory, better known as recursion theory, which plays a vital role in computer science.

# A Beginner's Guide to Mathematical Logic Dover Books on Mathematics

Dover Publications #ad - 2013 edition. Combining stories of great writers and philosophers with quotations and riddles, this completely original text for first courses in mathematical logic examines problems related to proofs, propositional logic and first-order logic, undecidability, and other topics.# Introduction to Mathematical Philosophy

Neeland Media LLC #ad - The work focuses on mathematical logic as related to traditional and contemporary philosophy, of which Russell remarks, "logic is the youth of mathematics and mathematics is the manhood of logic. It is regarded today as a lucid, accessible exploration of the gray area where mathematics and philosophy. He wrote his "introduction to mathematical Philosophy" 1919 in order to elucidate in a less technical way the main ideas of his and N.A. Whitehead's earlier "Principia Mathematica". As a mathematician, pacifist and social critic, bertrand russell is noted for his "revolt against idealism" in Britain in the early 20th century, historian, philosopher, socialist, as well as his pacifist activism during WWI, logician, a campaign against Adolf Hitler and later the United States' involvement in the Vietnam War.

Introduction to Mathematical Philosophy #ad - In addition to his political activism, he is considered to be one of the founders of analytic philosophy, receiving the Nobel Prize in Literature in 1950 for his various humanitarian and philosophical works.

# When Einstein Walked with Gödel: Excursions to the Edge of Thought

Farrar, Straus and Giroux #ad - Holt offers a painless and playful introduction to many of our most beautiful but least understood ideas, from Einsteinian relativity to string theory, and also invites us to consider why the greatest logician of the twentieth century believed the U. S. With his trademark clarity and humor, the quest for the foundations of mathematics, Holt probes the mysteries of quantum mechanics, and the nature of logic and truth.From jim holt, the new york times bestselling author of why does the World Exist?, comes an entertaining and accessible guide to the most profound scientific and mathematical ideas of recent centuries in When Einstein Walked with Gödel: Excursions to the Edge of Thought. Does time exist? what is infinity? why do mirrors reverse left and right but not up and down? In this scintillating collection, the cosmos, Holt explores the human mind, and the thinkers who’ve tried to encompass the latter with the former.

When Einstein Walked with Gödel: Excursions to the Edge of Thought #ad - Constitution contained a terrible contradiction—and whether the universe truly has a future. Along the way, he offers intimate biographical sketches of celebrated and neglected thinkers, from the physicist Emmy Noether to the computing pioneer Alan Turing and the discoverer of fractals, Benoit Mandelbrot.

# The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise Dover Books on Mathematics

Dover Publications #ad - Philosophers with only a basic grounding in mathematics, as well as mathematicians who have taken only an introductory course in philosophy, will find an abundance of intriguing topics in this text, which is appropriate for undergraduate-and graduate-level courses. She concludes with views of the constructs and reality of mathematical structure.Author mary tiles further examines permutations, combinations, and infinite cardinalities; numbering the continuum; Cantor's transfinite paradise; axiomatic set theory; logical objects and logical types; and independence results and the universe of sets. As cantor's sometime collaborator, remarked, David Hilbert, "No one will drive us from the paradise that Cantor has created.

The Philosophy of Set Theory: An Historical Introduction to Cantor's Paradise Dover Books on Mathematics #ad - This volume offers a guided tour of modern mathematics' Garden of Eden, beginning with perspectives on the finite universe and classes and Aristotelian logic. Cantor's innovation was opposed, by the establishment; years later, the value of his work was recognized and appreciated as a landmark in mathematical thought, and ignored, forming the beginning of set theory and the foundation for most of contemporary mathematics.

His methods, unorthodox for the time, enabled him to derive theorems that established a mathematical reality for a hierarchy of infinities. A century ago, georg Cantor demonstrated the possibility of a series of transfinite infinite numbers.