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The starting point for this monograph is the previously unknown
connection between the Continuum Hypothesis and the saturation of
the non-stationary ideal on 1; and the principle result of this
monograph is the identification of a canonical model in which the
Continuum Hypothesis is false. This is the first example of such a
model and moreover the model can be characterized in terms of
maximality principles concerning the universal-existential theory
of all sets of countable ordinals. This model is arguably the long
sought goal of the study of forcing axioms and iterated forcing but
is obtained by completely different methods, for example no theory
of iterated forcing whatsoever is required. The construction of the
model reveals a powerful technique for obtaining independence
results regarding the combinatorics of the continuum, yielding a
number of results which have yet to be obtained by any other
method. This monograph is directed to researchers and advanced
graduate students in Set Theory. The second edition is updated to
take into account some of the developments in the decade since the
first edition appeared, this includes a revised discussion of
-logic and related matters.
The infinite No other question has ever moved so profoundly the
spirit of man; no other idea has so fruitfully stimulated his
intellect; yet no other concept stands in greater need of
clarification than that of the infinite. David Hilbert This
interdisciplinary study of infinity explores the concept through
the prism of mathematics and then offers more expansive
investigations in areas beyond mathematical boundaries to reflect
the broader, deeper implications of infinity for human intellectual
thought. More than a dozen world renowned researchers in the fields
of mathematics, physics, cosmology, philosophy, and theology offer
a rich intellectual exchange among various current viewpoints,
rather than displaying a static picture of accepted views on
infinity. The book starts with a historical examination of the
transformation of infinity from a philosophical and theological
study to one dominated by mathematics. It then offers technical
discussions on the understanding of mathematical infinity.
Following this, the book considers the perspectives of physics and
cosmology: Can infinity be found in the real universe? Finally, the
book returns to questions of philosophical and theological aspects
of infinity."
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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.
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.
This interdisciplinary study of infinity explores the concept
through the prism of mathematics and then offers more expansive
investigations in areas beyond mathematical boundaries to reflect
the broader, deeper implications of infinity for human intellectual
thought. More than a dozen world-renowned researchers in the fields
of mathematics, physics, cosmology, philosophy and theology offer a
rich intellectual exchange among various current viewpoints, rather
than displaying a static picture of accepted views on infinity. The
book starts with a historical examination of the transformation of
infinity from a philosophical and theological study to one
dominated by mathematics. It then offers technical discussions on
the understanding of mathematical infinity. Following this, the
book considers the perspectives of physics and cosmology: can
infinity be found in the real universe? Finally, the book returns
to questions of philosophical and theological aspects of infinity.
Super-real fields are a class of large totally ordered fields.
These fields are larger than the real line. They arise from
quotients of the algebra of continuous functions on a compact space
by a prime ideal, and generalize the well-known class of
ultrapowers, and indeed the continuous ultrapowers. These fields
are of interest in their own right and have many surprising
applications, both in analysis and logic. The authors introduce
some exciting new fields, including a natural generalization of the
real line R, and resolve a number of open problems. The book is
intended to be accessible to analysts and logicians. After an
exposition of the general theory of ordered fields and a careful
proof of some classic theorems, including Kaplansky's embedding
theorems , the authors establish important new results in Banach
algebra theory, non-standard analysis, an model theory.
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R398
R330
Discovery Miles 3 300
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