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Set Theory for the Working Mathematician

Author: Krzysztof Ciesielski

Publisher: Cambridge University Press

ISBN:

Category: Mathematics

Page: 236

View: 502

Presents those methods of modern set theory most applicable to other areas of pure mathematics.

A First Course in Mathematical Logic and Set Theory

Author: Michael L. O'Leary

Publisher: John Wiley & Sons

ISBN:

Category: Mathematics

Page: 464

View: 327

Rather than teach mathematics and the structure of proofssimultaneously, this book first introduces logic as the foundationof proofs and then demonstrates how logic applies to mathematicaltopics. This method ensures that readers gain a firmunderstanding of how logic interacts with mathematics and empowersthem to solve more complex problems. The study of logic andapplications is used throughout to prepare readers for further workin proof writing. Readers are first introduced tomathematical proof-writing, and then the book provides anoverview of symbolic logic that includes two-column logicproofs. Readers are then transitioned to set theory andinduction, and applications of number theory, relations, functions,groups, and topology are provided to further aid incomprehension. Topical coverage includes propositional logic,predicate logic, set theory, mathematical induction, number theory,relations, functions, group theory, and topology.

Introduction to Model Theory

Author: Philipp Rothmaler

Publisher: CRC Press

ISBN:

Category: Mathematics

Page: 324

View: 686

Model theory investigates mathematical structures by means of formal languages. So-called first-order languages have proved particularly useful in this respect. This text introduces the model theory of first-order logic, avoiding syntactical issues not too relevant to model theory. In this spirit, the compactness theorem is proved via the algebraically useful ultrsproduct technique (rather than via the completeness theorem of first-order logic). This leads fairly quickly to algebraic applications, like Malcev's local theorems of group theory and, after a little more preparation, to Hilbert's Nullstellensatz of field theory. Steinitz dimension theory for field extensions is obtained as a special case of a much more general model-theoretic treatment of strongly minimal theories. There is a final chapter on the models of the first-order theory of the integers as an abelian group. Both these topics appear here for the first time in a textbook at the introductory level, and are used to give hints to further reading and to recent developments in the field, such as stability (or classification) theory.

The Foundations of Mathematics

Author: Ian Stewart

Publisher: OUP Oxford

ISBN:

Category: Mathematics

Page: 432

View: 505

The transition from school mathematics to university mathematics is seldom straightforward. Students are faced with a disconnect between the algorithmic and informal attitude to mathematics at school, versus a new emphasis on proof, based on logic, and a more abstract development of general concepts, based on set theory. The authors have many years' experience of the potential difficulties involved, through teaching first-year undergraduates and researching the ways in which students and mathematicians think. The book explains the motivation behind abstract foundational material based on students' experiences of school mathematics, and explicitly suggests ways students can make sense of formal ideas. This second edition takes a significant step forward by not only making the transition from intuitive to formal methods, but also by reversing the process- using structure theorems to prove that formal systems have visual and symbolic interpretations that enhance mathematical thinking. This is exemplified by a new chapter on the theory of groups. While the first edition extended counting to infinite cardinal numbers, the second also extends the real numbers rigorously to larger ordered fields. This links intuitive ideas in calculus to the formal epsilon-delta methods of analysis. The approach here is not the conventional one of 'nonstandard analysis', but a simpler, graphically based treatment which makes the notion of an infinitesimal natural and straightforward. This allows a further vision of the wider world of mathematical thinking in which formal definitions and proof lead to amazing new ways of defining, proving, visualising and symbolising mathematics beyond previous expectations.

The British National Bibliography

Author: Arthur James Wells

Publisher:

ISBN:

Category: English literature

Page:

View: 640

Forthcoming Books

Author: Rose Arny

Publisher:

ISBN:

Category: American literature

Page:

View: 749

Clifford Algebras: An Introduction

Author: D. J. H. Garling

Publisher: Cambridge University Press

ISBN:

Category: Mathematics

Page: 200

View: 446

A straightforward introduction to Clifford algebras, providing the necessary background material and many applications in mathematics and physics.

The Bulletin of Mathematics Books

Author:

Publisher:

ISBN:

Category: Mathematics

Page:

View: 928

Elements of the Representation Theory of Associative Algebras: Volume 1

Techniques of Representation Theory

Author: Ibrahim Assem

Publisher: Cambridge University Press

ISBN:

Category: Mathematics

Page: 472

View: 107

Provides an elementary but up-to-date introduction to the representation theory of algebras.

Fourier Analysis on Finite Groups and Applications

Author: Audrey Terras

Publisher: Cambridge University Press

ISBN:

Category: Mathematics

Page: 442

View: 280

A friendly introduction to Fourier analysis on finite groups, accessible to undergraduates/graduates in mathematics, engineering and the physical sciences.

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