Table Of ContentKurt Gödel (1906-1978) was an Austrian-American mathematician, who is best
C
h
known for his incompleteness theorems. He was the greatest mathematical a
logician of the 20th century, with his contributions extending to Einstein’s itin
,
general relativity, as he proved that Einstein’s theory allows for time machines. d
a
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o
The Gödel incompleteness phenomenon - the usual formal mathematical systems s
ta
cannot prove nor disprove all true mathematical sentences - is frequently presented ,
D
in textbooks as something that happens in the rarefied realms of mathematical o
r
logic, and that has nothing to do with the real world. Practice shows the contrary ia
though; one can demonstrate the validity of the phenomenon in various areas,
ranging from chaos theory and physics to economics, and even ecology. In this
lively treatise, based on Chaitin’s groundbreaking work and on the da Costa-
Doria results in physics, ecology, economics and computer science, the authors
show that the Gödel incompleteness phenomenon can directly bear on the
G
practice of science and perhaps on our everyday life.
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d
This accessible book gives a new, detailed and elementary explanation of the
e
Gödel incompleteness theorems and presents the Chaitin results and their l
’
relation to the da Costa-Doria results, which are given in full, but with no s
technicalities. Besides theory, the historical report and personal stories about W
the main character and on this book’s writing process, make it appealing
a
leisure reading for those interested in mathematics, logic, physics, philosophy y
and computer sciences.
Gregory Chaitin is an Argentinian-American mathematician and
computer scientist. The author of many books and scholarly papers, Chaitin Gödel’s Way
proved the Gödel-Chaitin incompleteness theorem and is the discoverer of
the remarkable Omega number, which shows that God plays dice in pure
mathematics. Newton da Costa is a Brazilian logician whose best known
Exploits into an
contributions have been in the realms of nonclassical logics and philosophy of
science. Da Costa developed paraconsistent logics, that is, logical systems that
undecidable world
admit inner contradictions. Francisco Antonio Doria is a Brazilian physicist.
He has made contributions to the gauge field copy problem in quantum field
theory and proved with Newton da Costa several incompleteness theorems
in mathematics, physics and mathematical economics, including the
undecidability of chaos theory.
Gregory Chaitin
Newton da Costa
Francisco Antonio Doria
an informa business
G¨odel’s Way
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Apersonalaccountbysomeoftheparticipantsintheworkgoing
beyondGo¨delbyfindinguncomputabilityandincompletenessin
manyareasofcontinuousanddiscretemathematicsandtheoreti-
calphysics.
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G¨odel’s Way
Exploits into an undecidable world
Gregory Chaitin,
Newton da Costa &
Francisco Antonio Doria
CRC Press
Taylor & Francis Group
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© 2011 by Taylor & Francis Group, LLC
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Version Date: 20120127
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Contents
Prologue xiii
Acknowledgments xvii
AbouttheAuthors xix
ACaveat xxi
1.Go¨del,Turing 1
Go¨del:logicandtime 2
Ashortbiography 4
Theincompletenesstheorems,I 5
Kleene’sversionofthefirstincompletenesstheorem 6
AnimmediateconsequenceofKleene’sproof 7
The incompleteness theorems II: consistency cannot be proved
withinthesystem 8
Aweirdformalsystem 9
Canweprovetheconsistencyofarithmetic? 10
Chaitin’sincompletenesstheorem 11
Berry’sParadox 12
Rice’stheorem 13
MoreworkbyGo¨del:theconstructiveuniverseofsets 14
Aconcludingnote:Go¨delontimemachines 17
AlanTuringandhismathematicalmachines 20
Whatisacomputation? 20
Turingmachines,I 21
Turingmachines,II 22
Theuniversalmachine 23
Thehaltingproblem 23
Go¨del’sfirstincompletenesstheoremrevisited 24
TheChurch–Turingthesis 24
Diophantineequations;Hilbert’s10thproblem 25
Undecidableissues 27
ix
