UID:
almahu_9947367870002882
Umfang:
1 online resource (349 p.)
ISBN:
0-444-53745-7
,
9786612967450
,
1-282-96745-2
,
9786612035036
,
1-282-03503-7
,
978-008-096-1
Serie:
Studies in logic and the foundations of mathematics ; v. 106
Inhalt:
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 r
Anmerkung:
Includes indexes.
,
Front Cover; Combinatorial Set Theory: Partition Relations for Cardinals; Copyright Page; Contents; Preface; Chapter I. Introduction; 1. Notation and basic concepts; 2. The axioms of Zermelo-Fraenkel set theory; 3. Ordinals cardinals, and order types; 4. Basic tools of set theory; Chapter II. Preliminaries; 5. Stationary sets; 6. Equalities and inequalities for cardinals; 7. The logarithm operation; Chapter III. Fundamentals about partition relations; 8. A guide to partition symbols; 9. Elementary properties of the ordinary partition symbol; 10. Ramsey's theorem
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11. The Erdös-Dushnik-Miller theorem12. Negative relations with infinite superscripts; Chapter IV. Trees and positive ordinary partition relations; 13.Trees; 14. Tree arguments; 15. End-homogeneous sets; 16. The Stepping-up Lemma; 17. The main results in case r = 2 and k is regular; and some corollaries for r G 3; 18. A direct construction of the canonical partition tree; Chapter V. Negative ordinary partition relations, and the discussion of the finite case; 19. Multiplication of negative partition relations for r = 2; 20. A negative partition relation established with the aid of GCH
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21. Addition of negative partition relations for r =222. Addition of negative partition relations for r G 3; 23. Multiplication of negative partition relations in case r G 3; 24. The Negative Stepping-up Lemma; 25. Some special negative partition relations for r G 3; 26. The finite case of the ordinary partition relation; Chapter VI. The canonization lemmas; 27. Shelah's canonization; 28. The General Canonization Lemma; Chapter VII. Large cardinals; 29. The ordinary partition relation for inaccessible cardinals; 30. Weak compactness and a metamathematical approach to the Hanf-Tarski result
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31. Baumgartner's principle32. A combinatorial approach to the Hanf-Tarski result; 33. Hanf's iteration scheme; 34. Saturated ideals, measurable cardinals. and strong partition relations; Chapter VIII. Discussion of the ordinary partition relation with superscript 2; 35. Discussion of the ordinary partition symbol in case r = 2; 36. Discussion of the ordinary partition relation in case r=2 under the assumption of GCH; 37. Sierpinski partitions; Chapter IX. Discussion of the ordinary partition relation with superscript G 3; 38. Reduction of the superscript
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39. Applicability of the Reduction Theorem40. Consequences of the Reduction Theorem; 41. The main result for the case r G 3; 42. The main result for the case r G 3 with GCH; Chapter X. Some applications of combinatorial metbods; 43. Applications in topology; 44. Fodor's and Hajnal's set-mapping theorems; 45. Set mapping of type 〉 1; 46. Finite free sets with respect to set mappings of type 〉 1; 47. Inequalities for powers of singular cardinals; 48. Cardinal exponentiation and saturated ideals; Chapter XI. A brief survey of the square bracket relation
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49. Negative square bracket relations and the GCH
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English
Weitere Ausg.:
ISBN 963-05-2877-0
Weitere Ausg.:
ISBN 0-444-86157-2
Sprache:
Englisch
