|
Showing 1 - 10 of
10 matches in All Departments
An introduction to nonstandard analysis based on a course given by
the author. It is suitable for beginning graduates or upper
undergraduates, or for self-study by anyone familiar with
elementary real analysis. It presents nonstandard analysis not just
as a theory about infinitely small and large numbers, but as a
radically different way of viewing many standard mathematical
concepts and constructions. It is a source of new ideas, objects
and proofs, and a wealth of powerful new principles of reasoning.
The book begins with the ultrapower construction of hyperreal
number systems, and proceeds to develop one-variable calculus,
analysis and topology from the nonstandard perspective. It then
sets out the theory of enlargements of fragments of the
mathematical universe, providing a foundation for the full-scale
development of the nonstandard methodology. The final chapters
apply this to a number of topics, including Loeb measure theory and
its relation to Lebesgue measure on the real line. Highlights
include an early introduction of the ideas of internal, external
and hyperfinite sets, and a more axiomatic set-theoretic approach
to enlargements than is usual.
A classic exposition of a branch of mathematical logic that uses
category theory, this text is suitable for advanced undergraduates
and graduate students and accessible to both philosophically and
mathematically oriented readers. Robert Goldblatt is Professor of
Pure Mathematics at New Zealand's Victoria University. 1983
edition.
An introduction to nonstandard analysis based on a course given by the author. It is suitable for beginning graduates or upper undergraduates, or for self-study by anyone familiar with elementary real analysis. It presents nonstandard analysis not just as a theory about infinitely small and large numbers, but as a radically different way of viewing many standard mathematical concepts and constructions. It is a source of new ideas, objects and proofs, and a wealth of powerful new principles of reasoning. The book begins with the ultrapower construction of hyperreal number systems, and proceeds to develop one-variable calculus, analysis and topology from the nonstandard perspective. It then sets out the theory of enlargements of fragments of the mathematical universe, providing a foundation for the full-scale development of the nonstandard methodology. The final chapters apply this to a number of topics, including Loeb measure theory and its relation to Lebesgue measure on the real line. Highlights include an early introduction of the ideas of internal, external and hyperfinite sets, and a more axiomatic set-theoretic approach to enlargements than is usual.
This book examines the geometrical notion of orthogonality, and
shows how to use it as the primitive concept on which to base a
metric structure in affine geometry. The subject has a long
history, and an extensive literature, but whatever novelty there
may be in the study presented here comes from its focus on
geometries hav ing lines that are self-orthogonal, or even singular
(orthogonal to all lines). The most significant examples concern
four-dimensional special-relativistic spacetime (Minkowskian
geometry), and its var ious sub-geometries, and these will be
prominent throughout. But the project is intended as an exercise in
the foundations of geome try that does not presume a knowledge of
physics, and so, in order to provide the appropriate intuitive
background, an initial chapter has been included that gives a
description of the different types of line (timelike, spacelike,
lightlike) that occur in spacetime, and the physical meaning of the
orthogonality relations that hold between them. The
coordinatisation of affine spaces makes use of constructions from
projective geometry, including standard results about the ma trix
represent ability of certain projective transformations (involu
tions, polarities). I have tried to make the work sufficiently self
contained that it may be used as the basis for a course at the ad
vanced undergraduate level, assuming only an elementary knowledge
of linear and abstract algebra."
Advances in Modal Logic is a unique international forum for
presenting the latest results and new directions of research in
Modal Logic broadly conceived. The topics dealt with are of
interdisciplinary interest and range from mathematical,
computational, and philosophical problems to applications in
knowledge representation and formal linguistics. This volume
contains invited and contributed papers from the seventh conference
in the AiML series, held in Nancy, France, in September 2008. It
reports on substantial advances, both in the foundations of modal
logic and in a number of application areas. It includes papers on
the metatheory of a variety of modal logics; on systems for spatial
and temporal reasoning and interpreting natural language; on the
emerging coalgebraic perspective; and on historical views of the
nature of modality.
The Asian Logic Conference is the most significant logic meeting
outside of North America and Europe, and this volume represents
work presented at, and arising from the 12th meeting. It collects a
number of interesting papers from experts in the field. It covers
many areas of logic.
Many systems of quantified modal logic cannot be characterised by
Kripke's well-known possible worlds semantic analysis. This book
shows how they can be characterised by a more general 'admissible
semantics', using models in which there is a restriction on which
sets of worlds count as propositions. This requires a new
interpretation of quantifiers that takes into account the
admissibility of propositions. The author sheds new light on the
celebrated Barcan Formula, whose role becomes that of legitimising
the Kripkean interpretation of quantification. The theory is worked
out for systems with quantifiers ranging over actual objects, and
over all possibilia, and for logics with existence and identity
predicates and definite descriptions. The final chapter develops a
new admissible 'cover semantics' for propositional and quantified
relevant logic, adapting ideas from the Kripke Joyal semantics for
intuitionistic logic in topos theory. This book is for mathematical
or philosophical logicians, computer scientists and linguists.
Now revised and significantly expanded, this textbook introduces
modal logic and examines the relevance of modal systems for
theoretical computer science. Goldblatt sets out a basic theory of
normal modal and temporal propositional logics, including issues
such as completeness proofs, decidability, first-order
definability, and canonicity. The basic theory is then applied to
logics of discrete, dense, and continuous time; to the temporal
logic of concurrent programs involving the connectives henceforth,
next, and until; and to the dynamic logic of regular programs. New
material for the second edition extends the temporal logic of
concurrency to branching time, studying a system of Computational
Tree Logic that formalizes reasoning about behavior. Dynamic logic
is also extended to the case of concurrency, introducing a
connective for the parallel execution of commands. A separate
section is devoted to quantificational dynamic logic. Numerous
exercises are included for use in the classroom.Robert Goldblatt is
a professor of pure mathematics at the Victoria University of
Wellington, New Zealand.
Modal logic is the study of modalities - expressions that qualify
assertions about the truth of statements - like the ordinary
language phrases necessarily, possibly, it is known/believed/ought
to be, etc., and computationally or mathematically motivated
expressions like provably, at the next state, or after the
computation terminates. The study of modalities dates from
antiquity, but has been most actively pursued in the last three
decades, since the introduction of the methods of Kripke semantics,
and now impacts on a wide range of disciplines, including the
philosophy of language and linguistics ('possible words' semantics
for natural language), constructive mathematics (intuitionistic
logic), theoretical computer science (dynamic logic, temporal and
other logics for concurrency), and category theory (sheaf
semantics). This volume collects together a number of the author's
papers on modal logic, beginning with his work on the duality
between algebraic and set-theoretic modals, and including two new
articles, one on infinitary rules of inference, and the other about
recent results on the relationship between modal logic and
first-order logic. Another paper on the 'Henkin method' in
completeness proofs has been substantially extended to give new
applications. Additional articles are concerned with quantum logic,
provability logic, the temporal logic of relativistic spacetime,
modalities in topos theory, and the logic of programs.
Modal logic is the study of modalities - expressions that qualify
assertions about the truth of statements - like the ordinary
language phrases necessarily, possibly, it is known/believed/ought
to be, etc., and computationally or mathematically motivated
expressions like provably, at the next state, or after the
computation terminates. The study of modalities dates from
antiquity, but has been most actively pursued in the last three
decades, since the introduction of the methods of Kripke semantics,
and now impacts on a wide range of disciplines, including the
philosophy of language and linguistics ('possible words' semantics
for natural language), constructive mathematics (intuitionistic
logic), theoretical computer science (dynamic logic, temporal and
other logics for concurrency), and category theory (sheaf
semantics). This volume collects together a number of the author's
papers on modal logic, beginning with his work on the duality
between algebraic and set-theoretic modals, and including two new
articles, one on infinitary rules of inference, and the other about
recent results on the relationship between modal logic and
first-order logic. Another paper on the 'Henkin method' in
completeness proofs has been substantially extended to give new
applications. Additional articles are concerned with quantum logic,
provability logic, the temporal logic of relativistic spacetime,
modalities in topos theory, and the logic of programs.
|
You may like...
Southpaw
Jake Gyllenhaal, Forest Whitaker, …
DVD
R99
R24
Discovery Miles 240
|