Problems and Theorems in Classical Set TheorySpringer Science & Business Media, 22.11.2006 - 516 Seiten 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. |
Inhalt
3 | |
9 | |
13 | |
15 | |
19 | |
Ordered sets | 23 |
Order types 33 | 35 |
Ordinals | 36 |
Equivalence 159 | 149 |
Continuum | 163 |
Sets of reals and real functions | 173 |
Ordered sets | 185 |
Order types | 213 |
Ordinals | 223 |
Ordinal arithmetic 237 | 236 |
Cardinals | 265 |
Ordinal arithmetic | 43 |
Cardinals | 51 |
Partially ordered sets | 55 |
Transfinite enumeration 59 | 58 |
Euclidean spaces | 63 |
Zorns lemma | 65 |
Hamel bases | 67 |
Ultrafilters on ω | 75 |
Families of sets | 79 |
The BanachTarski paradox | 81 |
Stationary sets in ω1 85 | 84 |
Stationary sets in larger cardinals | 89 |
Canonical functions | 93 |
Infinite graphs | 95 |
Partition relations 101 | 100 |
Δsystems | 107 |
Set mappings | 109 |
Trees | 111 |
The measure problem 117 | 116 |
Stationary sets in λκ | 123 |
The axiom of choice 127 | 126 |
Wellfounded sets and the axiom of foundation | 129 |
Solutions | 133 |
Operations on sets | 135 |
Countability | 147 |
Partially ordered sets 275 | 274 |
Transfinite enumeration | 285 |
Euclidean spaces | 299 |
Zorns lemma 309 | 300 |
Hamel bases | 317 |
The continuum hypothesis | 327 |
Ultrafilters on ω | 341 |
Families of sets 351 | 350 |
The BanachTarski paradox | 359 |
Stationary sets in ω1 | 369 |
Stationary sets in larger cardinals | 377 |
Canonical functions 385 | 384 |
Infinite graphs | 389 |
Partition relations 405 | 404 |
Δsystems | 421 |
Set mappings | 427 |
Trees | 433 |
The measure problem 453 | 452 |
Stationary sets in λκ | 463 |
The axiom of choice | 471 |
Wellfounded sets and the axiom of foundation | 481 |
Glossary of Concepts | 493 |
Glossary of Symbols | 507 |
Andere Ausgaben - Alle anzeigen
Problems and Theorems in Classical Set Theory Peter Komjath,Vilmos Totik Eingeschränkte Leseprobe - 2006 |
Problems and Theorems in Classical Set Theory Peter Komjath,Vilmos Totik Keine Leseprobe verfügbar - 2010 |
Häufige Begriffe und Wortgruppen
0–1 sequences antichain arbitrary Assume axiom of choice Cardinal and Ordinal cardinality continuum cardinality smaller chromatic number claim clearly closed club set cofinal color contradiction countable set decomposed decomposition define densely ordered enumeration finite set finite subset function f graph Hamel basis hence implies increasing sequence induction hypothesis infinite cardinal infinite set intersection interval largest element lemma Let f limit ordinal Math natural numbers nonempty nonstationary normal form open sets order type Ordinal Numbers pairwise disjoint partially ordered set points Polish Sci power continuum preceding problem proof prove Publ rational numbers real numbers regular cardinal set of cardinality Sierpiński similar smallest element solution stationary set subgraph successor ordinal suppose Suslin tree theorem topology transfinite induction transfinite recursion tree ultrafilter unbounded set uncountable vertex vertices Warszawa well-ordered set