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This volume collects lecture notes from talks given in the Appalachian Set Theory workshop series (supported by the National Science Foundation) during the period 2006–2012.
This workshop series grew out of an informal series of expository lectures held at Carnegie Mellon University and attended by set theorists from universities in Appalachian states before 2006. The success of these earlier gatherings inspired the editors to formalize the series and seek funding to help more people attend. Participants from other universities were invited to host workshops as well. Typically there are three meetings a year with one taking place at Carnegie Mellon University and the remaining two elsewhere. Several of the workshops have been held in neighbouring regions but the series retains its Appalachian flavour.
At each workshop a leading researcher lectures for six hours on an important topic or technique in modern set theory. Students are engaged to assist in writing notes based on the lectures, and these notes are disseminated on the web. This provides a learning opportunity for the students and makes the notes universally available.
The papers collected here represent more polished versions of the lecture notes from most of the workshops to date. They were prepared collaboratively by the lecturers and the student assistants. The lecturers are the principal authors and their names appear first, followed by the names of the assistants. One workshop (represented in Chapter 7) had two lecturers, and two workshops (represented in Chapters 1 and 13) had two assistants.
This volume takes its name from a popular series of intensive mathematics workshops hosted at institutions in Appalachia and surrounding areas. At these meetings, internationally prominent set theorists give one-day lectures that focus on important new directions, methods, tools and results so that non-experts can begin to master these and incorporate them into their own research. Each chapter in this volume was written by the workshop leaders in collaboration with select student participants, and together they represent most of the meetings from the period 2006–2012. Topics covered include forcing and large cardinals, descriptive set theory, and applications of set theoretic ideas in group theory and analysis, making this volume essential reading for a wide range of researchers and graduate students.
In one sense, set theory is the study of mathematics using the tools of mathematics. After millennia of doing mathematics, mathematicians started trying to write down the rules of the game. Since mathematics had already fanned out into many subareas, each with its own terminology and concerns, the first task was to find a reasonable common language. It turns out that everything mathematicians do can be reduced to statements about sets, equality and membership. These three concepts are so fundamental that we cannot define them; we can only describe them. About equality alone, there is little to say other than “two things are equal if and only if they are the same thing.” Describing sets and membership has been trickier. After several decades and some false starts, mathematicians came up with a system of laws that reflected their intuition about sets, equality and membership, at least the intuition that they had built up so far. Most importantly, all of the theorems of mathematics that were known at the time could be derived from just these laws. In this context, it is common to refer to laws as axioms, and to this particular system as Zermelo–Fraenkel Set Theory with the Axiom of Choice, or ZFC. In the first unit of the course, through Chapter 4, we examine this system and get some practice using it to build up the theory of infinite numbers.
Set theory is the mathematics of infinity and part of the core curriculum for mathematics majors. This book blends theory and connections with other parts of mathematics so that readers can understand the place of set theory within the wider context. Beginning with the theoretical fundamentals, the author proceeds to illustrate applications to topology, analysis and combinatorics, as well as to pure set theory. Concepts such as Boolean algebras, trees, games, dense linear orderings, ideals, filters and club and stationary sets are also developed. Pitched specifically at undergraduate students, the approach is neither esoteric nor encyclopedic. The author, an experienced instructor, includes motivating examples and over 100 exercises designed for homework assignments, reviews and exams. It is appropriate for undergraduates as a course textbook or for self-study. Graduate students and researchers will also find it useful as a refresher or to solidify their understanding of basic set theory.