Mathematical foundations Books

Book SynopsisThe goal of this monograph is to give an accessible introduction to nonstandard methods and their applications, with an emphasis on combinatorics and Ramsey theory. It includes both new nonstandard proofs of classical results and recent developments initially obtained in the nonstandard setting. This makes it the first combinatorics-focused account of nonstandard methods to be aimed at a general (graduate-level) mathematical audience. This book will provide a natural starting point for researchers interested in approaching the rapidly growing literature on combinatorial results obtained via nonstandard methods. The primary audience consists of graduate students and specialists in logic and combinatorics who wish to pursue research at the interface between these areas.Table of Contents- Part I Preliminaries. - Ultrafilters. - Nonstandard Analysis. - Hyperfinite Generators of Ultrafilters. - Many Stars: Iterated Nonstandard Extensions. - LoebMeasure. - Part II Ramsey Theory. - Ramsey’s Theorem. - The Theorems of van der Waerden and Hales-Jewett. - From Hindman to Gowers. - Partition Regularity of Equations. - Part III Combinatorial Number Theory. - Densities and Structural Properties. - Working in the Remote Realm. - Jin’s Sumset Theorem. - Sumset Configurations in Sets of Positive Density. - Near Arithmetic Progressions in Sparse Sets. - The Interval Measure Property. - Part IV Other Topics. - Triangle Removal and Szemerédi Regularity. - Approximate Groups. - Foundations of Nonstandard Analysis.

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Book SynopsisThis undergraduate textbook promotes an active transition to higher mathematics. Problem solving is the heart and soul of this book: each problem is carefully chosen to demonstrate, elucidate, or extend a concept. More than 300 exercises engage the reader in extensive arguments and creative approaches, while exploring connections between fundamental mathematical topics.Divided into four parts, this book begins with a playful exploration of the building blocks of mathematics, such as definitions, axioms, and proofs. A study of the fundamental concepts of logic, sets, and functions follows, before focus turns to methods of proof. Having covered the core of a transition course, the author goes on to present a selection of advanced topics that offer opportunities for extension or further study. Throughout, appendices touch on historical perspectives, current trends, and open questions, showing mathematics as a vibrant and dynamic human enterprise.This second edition has been reorganized to better reflect the layout and curriculum of standard transition courses. It also features recent developments and improved appendices. An Invitation to Abstract Mathematics is ideal for those seeking a challenging and engaging transition to advanced mathematics, and will appeal to both undergraduates majoring in mathematics, as well as non-math majors interested in exploring higher-level concepts.From reviews of the first edition:Bajnok’s new book truly invites students to enjoy the beauty, power, and challenge of abstract mathematics. … The book can be used as a text for traditional transition or structure courses … but since Bajnok invites all students, not just mathematics majors, to enjoy the subject, he assumes very little background knowledge. Jill Dietz, MAA ReviewsThe style of writing is careful, but joyously enthusiastic…. The author’s clear attitude is that mathematics consists of problem solving, and that writing a proof falls into this category. Students of mathematics are, therefore, engaged in problem solving, and should be given problems to solve, rather than problems to imitate. The author attributes this approach to his Hungarian background … and encourages students to embrace the challenge in the same way an athlete engages in vigorous practice. John Perry, zbMATHTable of ContentsPreface to Instructors.- Preface to Students.- Acknowledgments.- I What's Mathematics.- 1 Let's Play a Game!.- 2 What's the Name of the Game?.- 3 How to Make a Statement.- 4 What's True in Mathematics?.- A Ten Famous Conjectures.-B Ten Famous Theorems.- II The Foundations of Mathematics.- 5 Let's Be Logical!.- 6 Setting Examples.- 7 Quantifier Mechanics.- 8 Let's Be Functional!.- C The Foundations of Set Theory.- III How to Prove It.- 9 Universal Proofs.- 10 The Domino Theory.- 11 More Domino Games.- 12 Existential Proofs.- D Ten Famous Problems.- IV Advanced Math for Beginners.- 13 Mathematical Structures.- 14 Working in the Fields (and Other Structures).- 15 Group Work.- 16 Good Relations.- 17 Order, Please!.- 18 Now That's the Limit!.- 19 Sizing It Up.- 20 Infinite Delights.- 21 Number Systems Systematically.- 22 Games Are Valuable!.- E Graphic Content.- F All Games Considered.- G A Top Forty List of Math Theorems.

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