An Introduction to Gödel's Theorems: Edition 2

Cambridge University Press
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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 - extensively rewritten for its second edition - 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 was formerly Senior Lecturer in Philosophy at the University of Cambridge. His books include Explaining Chaos (1998) and An Introduction to Formal Logic (2003) and he is also a former editor of the journal Analysis.

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Additional Information

Publisher
Cambridge University Press
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Published on
Feb 21, 2013
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Pages
406
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ISBN
9781107328488
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Language
English
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Genres
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 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?

How Not to Be Wrong presents the surprising revelations behind all of these questions and many more, using the mathematician’s method of analyzing life and exposing the hard-won insights of the academic community to the layman—minus the jargon. Ellenberg chases mathematical threads through a vast range of time and space, from the everyday to the cosmic, encountering, among other things, baseball, Reaganomics, daring lottery schemes, Voltaire, the replicability crisis in psychology, Italian Renaissance painting, artificial languages, the development of non-Euclidean geometry, the coming obesity apocalypse, Antonin Scalia’s views on crime and punishment, the psychology of slime molds, what Facebook can and can’t figure out about you, and the existence of God.

Ellenberg pulls from history as well as from the latest theoretical developments to provide those not trained in math with the knowledge they need. Math, as Ellenberg says, is “an atomic-powered prosthesis that you attach to your common sense, vastly multiplying its reach and strength.” With the tools of mathematics in hand, you can understand the world in a deeper, more meaningful way. How Not to Be Wrong will show you how.
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.

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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