An Introduction to Gödel's Theorems

Cambridge University Press
Free sample

In 1931, the young Kurt Gödel published his First Incompleteness Theorem, which tells us that, for any sufficiently rich theory of arithmetic, there are some arithmetical truths the theory cannot prove. This remarkable result is among the most intriguing (and most misunderstood) in logic. Gödel also outlined an equally significant Second Incompleteness Theorem. How are these Theorems established, and why do they matter? Peter Smith answers these questions by presenting an unusual variety of proofs for the First Theorem, showing how to prove the Second Theorem, and exploring a family of related results (including some not easily available elsewhere). The formal explanations are interwoven with discussions of the wider significance of the two Theorems. This book will be accessible to philosophy students with a limited formal background. It is equally suitable for mathematics students taking a first course in mathematical logic.
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About the author

Peter Smith is Lecturer in Philosophy at the University of Cambridge. His books include Explaining Chaos (1998) and An Introduction to Formal Logic (2003), and he is a former editor of the journal Analysis.

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Additional Information

Publisher
Cambridge University Press
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Published on
Jul 26, 2007
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Pages
376
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ISBN
9781139465939
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Best For
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Language
English
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Genres
Mathematics / General
Mathematics / History & Philosophy
Mathematics / Logic
Philosophy / General
Philosophy / Logic
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Content Protection
This content is DRM protected.
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The official book behind the Academy Award-winning film The Imitation Game, starring Benedict Cumberbatch and Keira Knightley

It is only a slight exaggeration to say that the British mathematician Alan Turing (1912-1954) saved the Allies from the Nazis, invented the computer and artificial intelligence, and anticipated gay liberation by decades--all before his suicide at age forty-one. This New York Times–bestselling biography of the founder of computer science, with a new preface by the author that addresses Turing's royal pardon in 2013, is the definitive account of an extraordinary mind and life.


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Most philosophical writing is impersonal and argumentative, but many important philosophers have nevertheless written accounts of their own lives.

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Instead of the common impersonal and argumentative forms of ordinary philosophical discussion, these autobiographical texts are deeply personal and largely narrative or explanatory. The contributors to this book examine the philosophical significance of philosophers’ autobiographies and whether or not there are broadly philosophical tasks for which this sort of writing is particularly suited. Autobiography as Philosophy contains a general discussion about the relation between philosophical and autobiographical writing, and essays on the specific writings of Augustine, Abelard, Montaigne, Descartes, Vico, Hume, Rousseau, Newman, Mill, Nietzsche, Collingwood and Russell by specialists on the works of these individuals.

Original and distinctive in its efforts to think about the writings of historically recognized philosophers as communicative acts governed by their own distinctive interests and purposes, the book reveals that it is as much about the texts and the authors as it is about their doctrines and arguments. As a result the book steps back from many of the issues of substantive philosophical discussion to reflect on certain forms of writing as means to philosophical ends, to consider what those ends have included.

This biography of Jacques Derrida (1930–2004) tells the story of a Jewish boy from Algiers, excluded from school at the age of twelve, who went on to become the most widely translated French philosopher in the world – a vulnerable, tormented man who, throughout his life, continued to see himself as unwelcome in the French university system. We are plunged into the different worlds in which Derrida lived and worked: pre-independence Algeria, the microcosm of the École Normale Supérieure, the cluster of structuralist thinkers, and the turbulent events of 1968 and after. We meet the remarkable series of leading writers and philosophers with whom Derrida struck up a friendship: Louis Althusser, Emmanuel Levinas, Jean Genet, and Hélène Cixous, among others. We also witness an equally long series of often brutal polemics fought over crucial issues with thinkers such as Michel Foucault, Jacques Lacan, John R. Searle, and Jürgen Habermas, as well as several controversies that went far beyond academia, the best known of which concerned Heidegger and Paul de Man. We follow a series of courageous political commitments in support of Nelson Mandela, illegal immigrants, and gay marriage. And we watch as a concept – deconstruction – takes wing and exerts an extraordinary influence way beyond the philosophical world, on literary studies, architecture, law, theology, feminism, queer theory, and postcolonial studies.

In writing this compelling and authoritative biography, Benoît Peeters talked to over a hundred individuals who knew and worked with Derrida. He is also the first person to make use of the huge personal archive built up by Derrida throughout his life and of his extensive correspondence. Peeters’ book gives us a new and deeper understanding of the man who will perhaps be seen as the major philosopher of the second half of the twentieth century.

The Freakonomics of math—a math-world superstar unveils the hidden beauty and logic of the world and puts its power in our hands

The math we learn in school can seem like a dull set of rules, laid down by the ancients and not to be questioned. In How Not to Be Wrong, Jordan Ellenberg shows us how terribly limiting this view is: Math isn’t confined to abstract incidents that never occur in real life, but rather touches everything we do—the whole world is shot through with it.

Math allows us to see the hidden structures underneath the messy and chaotic surface of our world. It’s a science of not being wrong, hammered out by centuries of hard work and argument. Armed with the tools of mathematics, we can see through to the true meaning of information we take for granted: How early should you get to the airport? What does “public opinion” really represent? Why do tall parents have shorter children? Who really won Florida in 2000? And how likely are you, really, to develop cancer?

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When this book was first published in 1975 it was at once
enthusiastically received by scholars and the general public alike and
recognized as a classic of its genre. It represented a notable publication of
the early fruits of the Commission's work on the side of its responsibility for
the National Monuments Record for Wales. During the years which have since
intervened, much fresh information has come to light concerning Welsh houses -
not least because of the intense interest awakened by the original publication.
This new knowledge has, as far as possible, been incorporated in the new and
revised edition, which contains approximately onequarter more material than the
first. Although it has not been possible to alter the original text, a number
of additional maps and photographs have been included and a new dust-jacket has
been designed. The Commissioners would wish warmly to congratulate their
Secretary, Mr. Peter Smith, those of his colleagues who were associated with
him, and H.M.S.O. on the excellence of this volume. It marks another
outstanding landmark in the study of vernacular architecture, not only in Wales
but also in the British Isles, and a major achievement on the part of its
author.

Although this second edition of Houses of the Welsh
Countryside retains in their entirety the text, the illustrations, and the
layout of the volume first published in 1975, it also includes a substantial
amount of new information which has come to light since that date. Some of this
new material takes the form of additional figures inserted where appropriate
into the existing illustrative pages. Similarly a small number of additional
colour plates showing typical houses in characteristic settings has been tipped
into the text. There are also additions to the original map lists. It has not
been possible for reasons of cost to bring the maps themselves up to date, but
as the newly-discovered sites nearly always reinforce the distribution patterns
first indicated, this omission is not crucial. The numbers of new discoveries
can vary from a mere handful on one list to several hundred on another. All
other new material is introduced as part of an additional SECTION IV at the
back of the volume.



This section comprises:

Corrigenda

Covering sites which were inadequately or incorrectly described in the first
volume, involving in one case a complete reappraisal of the original reference.



Addenda I

Describing and illustrating a small number of newly surveyed houses of especial
interest which could not easily be fitted into the illustrations in the main
text.



Addenda II

Analysing the incidence of date-inscriptions as evidence for building activity.



Addenda III

Listing and mapping a number of features of domestic architecture not
previously so noted.



Addenda IV

Listing and mapping various features of ecclesiastical architecture which also
occur in houses and which therefore have a bearing on the evolution of domestic
architecture.

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