Erdös / Máté / Hajnal Combinatorial Set Theory: Partition Relations for Cardinals

1. Auflage 2011
ISBN: 978-0-444-53745-4
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 106, 348 Seiten, Web PDF

Reihe: Studies in Logic and the Foundations of Mathematics

This work presents the most important combinatorial ideas in partition calculus and discusses ordinary partition relations for cardinals without the assumption of the generalized continuum hypothesis. A separate section of the book describes the main partition symbols scattered in the literature. A chapter on the applications of the combinatorial methods in partition calculus includes a section on topology with Arhangel'skii's famous result that a first countable compact Hausdorff space has cardinality, at most continuum. Several sections on set mappings are included as well as an account of recent inequalities for cardinal powers that were obtained in the wake of Silver's breakthrough result saying that the continuum hypothesis can not first fail at a singular cardinal of uncountable cofinality.

Autoren/Hrsg.

Erdös, P.

Máté, A.

Hajnal, A.

Rado, P.

Inhaltsverzeichnis

1;Front Cover;1
2;Combinatorial Set Theory: Partition Relations for Cardinals ;4
3;Copyright Page;5
4;Contents;8
5;Preface;6
6;Chapter I. Introduction;10
6.1;1. Notation and basic concepts;10
6.2;2. The axioms of Zermelo–Fraenkel set theory;14
6.3;3. Ordinals cardinals, and order types;17
6.4;4. Basic tools of set theory;22
7;Chapter II. Preliminaries;35
7.1;5. Stationary sets;35
7.2;6. Equalities and inequalities for cardinals;40
7.3;7. The logarithm operation;46
8;Chapter III. Fundamentals about partition relations;53
8.1;8. A guide to partition symbols;53
8.2;9. Elementary properties of the ordinary partition symbol;63
8.3;10. Ramsey’s theorem;66
8.4;11. The Erdös–Dushnik–Miller theorem;71
8.5;12. Negative relations with infinite superscripts;79
9;Chapter IV. Trees and positive ordinary partition relations;81
9.1;13.Trees;81
9.2;14. Tree arguments;83
9.3;15. End-homogeneous sets;87
9.4;16. The Stepping-up Lemma;91
9.5;17. The main results in case r = 2 and k is regular; and some corollaries for r G 3;94
9.6;18. A direct construction of the canonical partition tree;101
10;Chapter V. Negative ordinary partition relations, and the discussion of the finite case;106
10.1;19. Multiplication of negative partition relations for r = 2;106
10.2;20. A negative partition relation established with the aid of GCH;111
10.3;21. Addition of negative partition relations for r =2;114
10.4;22. Addition of negative partition relations for r G 3;118
10.5;23. Multiplication of negative partition relations in case r G 3;123
10.6;24. The Negative Stepping-up Lemma;133
10.7;25. Some special negative partition relations for r G 3;139
10.8;26. The finite case of the ordinary partition relation;146
11;Chapter VI. The canonization lemmas;159
11.1;27. Shelah's canonization;159
11.2;28. The General Canonization Lemma;164
12;Chapter VII. Large cardinals;169
12.1;29. The ordinary partition relation for inaccessible cardinals;169
12.2;30. Weak compactness and a metamathematical approach to the Hanf–Tarski result;178
12.3;31. Baumgartner's principle;189
12.4;32. A combinatorial approach to the Hanf-Tarski result;195
12.5;33. Hanf's iteration scheme;201
12.6;34. Saturated ideals, measurable cardinals. and strong partition relations;202
13;Chapter VIII. Discussion of the ordinary partition relation with superscript 2;216
13.1;35. Discussion of the ordinary partition symbol in case r = 2;216
13.2;36. Discussion of the ordinary partition relation in case r=2 under the assumption of GCH;227
13.3;37. Sierpinski partitions;230
14;Chapter IX. Discussion of the ordinary partition relation with superscript G 3;234
14.1;38. Reduction of the superscript;234
14.2;39. Applicability of the Reduction Theorem;241
14.3;40. Consequences of the Reduction Theorem;244
14.4;41. The main result for the case r G 3;254
14.5;42. The main result for the case r G 3 with GCH;260
15;Chapter X. Some applications of combinatorial metbods;264
15.1;43. Applications in topology;264
15.2;44. Fodor's and Hajnal's set-mapping theorems;273
15.3;45. Set mapping of type > 1 ;276
15.4;46. Finite free sets with respect to set mappings of type > 1;286
15.5;47. Inequalities for powers of singular cardinals;289
15.6;48. Cardinal exponentiation and saturated ideals;302
16;Chapter XI. A brief survey of the square bracket relation;314
16.1;49. Negative square bracket relations and the GCH;314
16.2;50. The effect of a Suslin tree on negative relations;319
16.3;51. The Kurepa Hypothesis and negative stepping up to superscript 3;322
16.4;52. Aronszajn trees and Specker types (preparations for results without GCH) ;325
16.5;53. Negative square bracket relations without GCH;328
16.6;54. Positive relations for singular strong limit cardinals;331
16.7;55. Infinitary Jónsson algebras;333
17;Bibliography;336
18;Author index;342
19;Subject index;344