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This monograph presents recursion theory from a generalized point
of view centered on the computational aspects of definability. A
major theme is the study of the structures of degrees arising from
two key notions of reducibility, the Turing degrees and the
hyperdegrees, using techniques and ideas from recursion theory,
hyperarithmetic theory, and descriptive set theory. The emphasis is
on the interplay between recursion theory and set theory, anchored
on the notion of definability. The monograph covers a number of
fundamental results in hyperarithmetic theory as well as some
recent results on the structure theory of Turing and hyperdegrees.
It also features a chapter on the applications of these
investigations to higher randomness.
This volume comprises a collection of twenty written versions of
invited as well as contributed papers presented at the conference
held from 20-24 May 1996 in Beijing, China. It covers many areas of
logic and the foundations of mathematics, as well as computer
science. Also included is an article by M. Yasugi on the Asian
Logic Conference which first appeared in Japanese, to provide a
glimpse into the history and development of the series.
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Infinity And Truth (Hardcover)
Chitat Chong, Qi Feng, Theodore a. Slaman, W. Hugh Woodin
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R1,682
R1,446
Discovery Miles 14 460
Save R236 (14%)
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Ships in 12 - 17 working days
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This volume is based on the talks given at the Workshop on Infinity
and Truth held at the Institute for Mathematical Sciences, National
University of Singapore, from 25 to 29 July 2011. The chapters
cover topics in mathematical and philosophical logic that examine
various aspects of the foundations of mathematics. The theme of the
volume focuses on two basic foundational questions: (i) What is the
nature of mathematical truth and how does one resolve questions
that are formally unsolvable within the Zermelo-Fraenkel Set Theory
with the Axiom of Choice, and (ii) Do the discoveries in
mathematics provide evidence favoring one philosophical view over
others? These issues are discussed from the vantage point of recent
progress in foundational studies.The final chapter features
questions proposed by the participants of the Workshop that will
drive foundational research. The wide range of topics covered here
will be of interest to students, researchers and mathematicians
concerned with issues in the foundations of mathematics.
This volume presents the lecture notes of short courses given by
three leading experts in mathematical logic at the 2010 and 2011
Asian Initiative for Infinity Logic Summer Schools. The major
topics covered set theory and recursion theory, with particular
emphasis on forcing, inner model theory and Turing degrees,
offering a wide overview of ideas and techniques introduced in
contemporary research in the field of mathematical logic.
Although the Fields Medal does not have the same public recognition
as the Nobel Prizes, they share a similar intellectual standing. It
is restricted to one field - that of mathematics. The medal is
awarded to the best mathematicians who are 40 or younger, every
four years.A list of Fields Medallists and their contributions
provides a bird's-eye view of the major developments in mathematics
over the past 80 years. It highlights the areas in which, at
various times, the greatest progress has been made.The third
edition of Fields Medallists' Lectures features additional
contributions from: John W Milnor (1962), Enrico Bombieri (1974),
Gerd Faltings (1986), Andrei Okounkov (2006), Terence Tao (2006),
Cedric Villani (2010), Elon Lindenstrauss (2010), Ngo Bao Chau
(2010), Stanislav Smirnov (2010).
Although the Fields Medal does not have the same public recognition
as the Nobel Prizes, they share a similar intellectual standing. It
is restricted to one field - that of mathematics. The medal is
awarded to the best mathematicians who are 40 or younger, every
four years.A list of Fields Medallists and their contributions
provides a bird's-eye view of the major developments in mathematics
over the past 80 years. It highlights the areas in which, at
various times, the greatest progress has been made.The third
edition of Fields Medallists' Lectures features additional
contributions from: John W Milnor (1962), Enrico Bombieri (1974),
Gerd Faltings (1986), Andrei Okounkov (2006), Terence Tao (2006),
Cedric Villani (2010), Elon Lindenstrauss (2010), Ngo Bao Chau
(2010), Stanislav Smirnov (2010).
This book is a brief and focused introduction to the reverse
mathematics and computability theory of combinatorial principles,
an area of research which has seen a particular surge of activity
in the last few years. It provides an overview of some fundamental
ideas and techniques, and enough context to make it possible for
students with at least a basic knowledge of computability theory
and proof theory to appreciate the exciting advances currently
happening in the area, and perhaps make contributions of their own.
It adopts a case-study approach, using the study of versions of
Ramsey's Theorem (for colorings of tuples of natural numbers) and
related principles as illustrations of various aspects of
computability theoretic and reverse mathematical analysis. This
book contains many exercises and open questions.
This volume presents the lecture notes of short courses given by
three leading experts in mathematical logic at the 2012 Asian
Initiative for Infinity Logic Summer School. The major topics cover
set-theoretic forcing, higher recursion theory, and applications of
set theory to C*-algebra. This volume offers a wide spectrum of
ideas and techniques introduced in contemporary research in the
field of mathematical logic to students, researchers and
mathematicians.
This volume presents the lecture notes of short courses given by
three leading experts in mathematical logic at the 2012 Asian
Initiative for Infinity Logic Summer School. The major topics cover
set-theoretic forcing, higher recursion theory, and applications of
set theory to C*-algebra. This volume offers a wide spectrum of
ideas and techniques introduced in contemporary research in the
field of mathematical logic to students, researchers and
mathematicians.
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Notes On Forcing Axioms (Hardcover)
Stevo Todorcevic; Edited by Chitat Chong, Qi Feng, Theodore a. Slaman, W. Hugh Woodin, …
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R2,090
Discovery Miles 20 900
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Ships in 10 - 15 working days
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In the mathematical practice, the Baire category method is a tool
for establishing the existence of a rich array of generic
structures. However, in mathematics, the Baire category method is
also behind a number of fundamental results such as the Open
Mapping Theorem or the Banach-Steinhaus Boundedness Principle. This
volume brings the Baire category method to another level of
sophistication via the internal version of the set-theoretic
forcing technique. It is the first systematic account of
applications of the higher forcing axioms with the stress on the
technique of building forcing notions rather than on the
relationship between different forcing axioms or their consistency
strengths.
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