Synopses & Reviews
This volume contains a variety of problems from classical set theory and represents the first comprehensive collection of such problems. Many of these problems are also related to other fields of mathematics, including algebra, combinatorics, topology and real analysis. Rather than using drill exercises, most problems are challenging and require work, wit, and inspiration. They vary in difficulty, and are organized in such a way that earlier problems help in the solution of later ones. For many of the problems, the authors also trace the history of the problems and then provide proper reference at the end of the solution.
Review
From the reviews: "The volume contains 1007 problems in (mostly combinatorial) set theory. As indicated by the authors, "most of classical set theory is covered, classical in the sense that independence methods are not used, but classical also in the sense that most results come from the period, say, 1920--1970. Many problems are also related to other fields of mathematics such as algebra, combinatorics, topology and real analysis." And indeed the topics covered include applications of Zorn's lemma, Euclidean spaces, Hamel bases, the Banach-Tarski paradox and the measure problem. The statement of the problems, which are distributed among 31 chapters, takes 132 pages, and the (fairly detailed) solutions (together with some references) another 357 pages. Some problems are elementary but most of them are challenging. For example, in Chapter 29 the reader is asked in Problem 1 to show that $[\lambda]^{<\kappa}$ is="" the="" union="" of="" $\kappa$="" bounded="" sets,="" and="" in="" problem="" 20="" to="" prove="" baumgartner's="" result="" that="" every="" closed="" unbounded="" subset="" of=""><\aleph_1}$ is="" of="" maximal="" cardinality="" $\aleph_2^{\aleph_0}$.="" this="" is="" a="" welcome="" addition="" to="" the="" literature,="" which="" should="" be="" useful="" to="" students="" and="" researchers="" alike."="" (pierre="" matet,="" mathematical="" reviews)="" "the="" book="" is="" well="" written="" and="" self="" contained,="" a="" choice="" collection="" of="" hundreds="" of="" tastefully="" selected="" problems="" related="" to="" classical="" set="" theory,="" a="" wealth="" of="" naturally="" arising,="" simply="" formulated="" problems="" ...="" .="" it="" is="" certainly="" available="" to="" students="" of="" mathematics="" major="" even="" in="" their="" undergraduate="" years.="" the="" solutions="" contain="" the="" right="" amount="" of="" details="" for="" the="" targeted="" readership.="" ...="" this="" is="" a="" unique="" book,="" an="" excellent="" source="" to="" review="" the="" fundamentals="" of="" classical="" set="" theory,="" learn="" new="" tricks,="" discover="" more="" and="" more="" on="" the="" field."="" (tamás="" erdélyi,="" journal="" of="" approximation="" theory,="" 2008)="">
Synopsis
Although the ?rst decades of the 20th century saw some strong debates on set theory and the foundation of mathematics, afterwards set theory has turned into a solid branch of mathematics, indeed, so solid, that it serves as the foundation of the whole building of mathematics. Later generations, honest to Hilbert s dictum, No one can chase us out of the paradise that Cantor has created for us proved countless deep and interesting theorems and also applied the methods of set theory to various problems in algebra, topology, in?nitary combinatorics, and real analysis. The invention of forcing produced a powerful, technically sophisticated tool for solving unsolvable problems. Still, most results of the pre-Cohen era can be digested with just the knowledge of a commonsense introduction to the topic. And it is a worthy e?ort, here we refer not just to usefulness, but, ?rst and foremost, to mathematical beauty. In this volume we o?er a collection of various problems in set theory. Most of classical set theory is covered, classical in the sense that independence methods are not used, but classical also in the sense that most results come fromtheperiod, say,1920 1970.Manyproblemsarealsorelatedtoother?elds of mathematics such as algebra, combinatorics, topology, and real analysis. We do not concentrate on the axiomatic framework, although some - pects, such as the axiom of foundation or the role of the axiom of choice, are elaborated."
Synopsis
This is the first comprehensive collection of problems in set theory. It contains well chosen sequences of exercises. Most of the problems are challenging and require work, wit, and inspiration. The book is destined to become a classic.
Table of Contents
Foreword.- Problems: Operations on sets.- Countability.- Equivalence.- Continuum.- Sets of reals and real functions.- Ordered sets.- Order types.- Ordinals.- Ordinal arithmetic.- Cardinals.- Partially ordered sets.- Transfinite enumeration.- Euclidean spaces.- Zorn's lemma.- Hamel bases.- The continuum hypothesis.- Ultrafilters on w.- Families of sets.- The Banach-Tarski paradox.- Stationary sets in w1.- Stationary sets in larger cardinals.- Canonical functions.- Infinite graphs.- Partition relations.- \triangle systems.- Set mappings.- Trees.- The measure problem.- Stationary sets.- The axiom of choice.- Well founded sets and the axiom of foundation.- Solutions: Operations on sets.- Countability.- Equivalence.- Continuum.- Sets of reals and real functions.- Ordered sets.- Order types.- Ordinals.- Ordinal arithmetic.- Cardinals.- Partially ordered sets.- Transfinite enumeration.- Euclidean spaces.- Zorn's lemma.- Hamel bases.- The continuum hypothesis.- Ultrafilters on w Families of sets The Banach-Tarski paradox Stationary sets in w1.- Stationary sets in larger cardinals.- Canonical functions.- Infinite graphs.- Partition relations.- \triangle-systems.- Set mappings.- Trees.- The measure problem.- Stationary sets.- The axiom of choice Well founded sets and the axiom of foundation.- Appendix.- Glossary of Concepts.- Glossary of Symbols.- Index.