Problems and Theorems in Classical Set Theory

2006
ISBN: 978-0-387-30293-5
Verlag: Springer

Buch, Englisch, 516 Seiten, Format (B × H): 160 mm x 241 mm, Gewicht: 2010 g

Reihe: Problem Books in Mathematics

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 ˆ of the axiom of choice, are elaborated.

Zielgruppe

Graduate

Autoren/Hrsg.

Totik, Vilmos

Komjath, Peter

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Allgemein Grundlagen der Mathematik
• Mathematik | Informatik Mathematik Operations Research Graphentheorie

Inhaltsverzeichnis

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 ?.- Families of sets.- The Banach-Tarski paradox.- Stationary sets in ?1.- Stationary sets in larger cardinals.- Canonical functions.- Infinite graphs.- Partition relations.- ?-systems.- Set mappings.- Trees.- The measure problem.- Stationary sets in [?]^- The Banach-Tarski paradox.- Stationary sets in ?1.- Stationary sets in larger cardinals.- Canonical functions.- Infinite graphs.- Partition relations.- ?-systems.- Set mappings.- Trees.- The measure problem.- Stationary sets in [?]^