Set theory:
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2003
|
| Ausgabe: | 3. millenium ed, rev. and expanded |
| Schriftenreihe: | Springer monographs in mathematics
|
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009959885&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke |
| Umfang: | XIII, 769 S. |
| ISBN: | 3540440852 9783540440857 |
Internformat
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| 100 | 1 | |a Jech, Thomas J. |d 1944- |e Verfasser |0 (DE-588)107482673 |4 aut | |
| 245 | 1 | 0 | |a Set theory |c Thomas Jech |
| 250 | |a 3. millenium ed, rev. and expanded | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2003 | |
| 300 | |a XIII, 769 S. | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 0 | |a Springer monographs in mathematics | |
| 500 | |a Hier auch später erschienene, unveränderte Nachdrucke | ||
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| 650 | 4 | |a Set theory | |
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Datensatz im Suchindex
| DE-BY-TUM_call_number | 0102 MAT 040f 2005 A 2181 |
|---|---|
| DE-BY-TUM_katkey | 1497846 |
| DE-BY-TUM_location | 01 |
| DE-BY-TUM_media_number | 040010525387 |
| _version_ | 1821933234827558912 |
| adam_text | Table of Contents
Part I. Basic Set Theory
1. Axioms of Set Theory 3
Axioms of Zermelo Praenkel. Why Axiomatic Set Theory? Language of Set
Theory, Formulas. Classes. Extensionality. Pairing. Separation Schema.
Union. Power Set. Infinity. Replacement Schema. Exercises. Historical Notes.
2. Ordinal Numbers 17
Linear and Partial Ordering. Well Ordering. Ordinal Numbers. Induction and
Recursion. Ordinal Arithmetic. Well Founded Relations. Exercises. Historical
Notes.
3. Cardinal Numbers 27
Cardinality. Alephs. The Canonical Well Ordering of a x a. Cofinality. Ex¬
ercises. Historical Notes.
4. Real Numbers 37
The Cardinality of the Continuum. The Ordering of R. Suslin s Problem. The
Topology of the Real Line. Borel Sets. Lebesgue Measure. The Baire Space.
Polish Spaces. Exercises. Historical Notes.
5. The Axiom of Choice and Cardinal Arithmetic 47
The Axiom of Choice. Using the Axiom of Choice in Mathematics. The Count¬
able Axiom of Choice. Cardinal Arithmetic. Infinite Sums and Products. The
Continuum Function. Cardinal Exponentiation. The Singular Cardinal Hy¬
pothesis. Exercises. Historical Notes.
6. The Axiom of Regularity 63
The Cumulative Hierarchy of Sets. e Induction. Well Founded Relations. The
Bernays Godel Axiomatic Set Theory. Exercises. Historical Notes.
7. Filters, Ultrafilters and Boolean Algebras 73
Filters and Ultrafilters. Ultrafilters on w. K Complete Filters and Ideals.
Boolean Algebras. Ideals and Filters on Boolean Algebras. Complete Boolean
Algebras. Complete and Regular Subalgebras. Saturation. Distributivity of
Complete Boolean Algebras. Exercises. Historical Notes.
X Table of Contents
8. Stationary Sets 91
Closed Unbounded Sets. Mahlo Cardinals. Normal Filters. Silver s Theo¬
rem. A Hierarchy of Stationary Sets. The Closed Unbounded Filter on PK(X).
Exercises. Historical Notes.
9. Combinatorial Set Theory 107
Partition Properties. Weakly Compact Cardinals. Trees. Almost Disjoint Sets
and Functions. The Tree Property and Weakly Compact Cardinals. Ramsey
Cardinals. Exercises. Historical Notes.
10. Measurable Cardinals 125
The Measure Problem. Measurable and Real Valued Measurable Cardinals.
Measurable Cardinals. Normal Measures. Strongly Compact and Supercom
pact Cardinals. Exercises. Historical Notes.
11. Borel and Analytic Sets 139
Borel Sets. Analytic Sets. The Suslin Operation A.. The Hierarchy of Projective
Sets. Lebesgue Measure. The Property of Baire. Analytic Sets: Measure,
Category, and the Perfect Set Property. Exercises. Historical Notes.
12. Models of Set Theory 155
Review of Model Theory. Godel s Theorems. Direct Limits of Models. Re¬
duced Products and Ultraproducts. Models of Set Theory and Relativization.
Relative Consistency. Transitive Models and Ao Formulas. Consistency of
the Axiom of Regularity. Inaccessibility of Inaccessible Cardinals. Reflection
Principle. Exercises. Historical Notes.
Part II. Advanced Set Theory
13. Constructible Sets 175
The Hierarchy of Constructible Sets. Godel Operations. Inner Models of ZF.
The Levy Hierarchy. Absoluteness of Constructibility. Consistency of the Ax¬
iom of Choice. Consistency of the Generalized Continuum Hypothesis. Relative
Constructibility. Ordinal Definable Sets. More on Inner Models. Exercises.
Historical Notes.
14. Forcing 201
Forcing Conditions and Generic Sets. Separative Quotients and Complete
Boolean Algebras. Boolean Valued Models. The Boolean Valued Model VB.
The Forcing Relation. The Forcing Theorem and the Generic Model Theorem.
Consistency Proofs. Independence of the Continuum Hypothesis. Indepen¬
dence of the Axiom of Choice. Exercises. Historical Notes.
15. Applications of Forcing 225
Cohen Reals. Adding Subsets of Regular Cardinals. The K Chain Condition.
Distributivity. Product Forcing. Easton s Theorem. Forcing with a Class of
Conditions. The Levy Collapse. Suslin Trees. Random Reals. Forcing with
Perfect Trees. More on Generic Extensions. Symmetric Submodels of Generic
Models. Exercises. Historical Notes.
Table of Contents XI
16. Iterated Forcing and Martin s Axiom 267
Two Step Iteration. Iteration with Finite Support. Martin s Axiom. Indepen¬
dence of Suslin s Hypothesis. More Applications of Martin s Axiom. Iterated
Forcing. Exercises. Historical Notes.
17. Large Cardinals 285
Ultrapowers and Elementary Embeddings. Weak Compactness. Indescribabil
ity. Partitions and Models. Exercises. Historical Notes.
18. Large Cardinals and L 311
Silver Indiscernibles. Models with Indiscernibles. Proof of Silver s Theorem
and 0 . Elementary Embeddings of L. Jensen s Covering Theorem. Exercises.
Historical Notes.
19. Iterated Ultrapowers and L[U] 339
The Model L[U]. Iterated Ultrapowers. Representation of Iterated Ultrapow¬
ers. Uniqueness of the Model L[D]. Indiscernibles for L[D . General Iterations.
The Mitchell Order. The Models L[U]. Exercises. Historical Notes.
20. Very Large Cardinals 365
Strongly Compact Cardinals. Supercompact Cardinals. Beyond Supercom
pactness. Extenders and Strong Cardinals. Exercises. Historical Notes.
21. Large Cardinals and Forcing 389
Mild Extensions. Kunen Paris Forcing. Silver s Forcing. Prikry Forcing. Mea
surability of Ni in ZF. Exercises. Historical Notes.
22. Saturated Ideals 409
Real Valued Measurable Cardinals. Generic Ultrapowers. Precipitous Ideals.
Saturated Ideals. Consistency Strength of Precipitousness. Exercises. Histor¬
ical Notes.
23. The Nonstationary Ideal 441
Some Combinatorial Principles. Stationary Sets in Generic Extensions. Pre¬
cipitousness of the Nonstationary Ideal. Saturation of the Nonstationary Ideal.
Reflection. Exercises. Historical Notes.
24. The Singular Cardinal Problem 457
The Galvin Hajnal Theorem. Ordinal Functions and Scales. The pcf Theory.
The Structure of pcf. Transitive Generators and Localization. Shelah s Bound
on 2* . Exercises. Historical Notes.
25. Descriptive Set Theory 479
The Hierarchy of Projective Sets, n} Sets. Trees, Well Founded Relations
and K Suslin Sets. £2 Sets. Projective Sets and Constructibility. Scales and
Uniformization. £2 Well Orderings and Eg Well Founded Relations. Borel
Codes. Exercises. Historical Notes.
XII Table of Contents
26. The Real Line 511
Random and Cohen reals. Solovay Sets of Reals. The Levy Collapse. Solo
vay s Theorem. Lebesgue Measurability of S2 Sets. Ramsey Sets of Reals and
Mathias Forcing. Measure and Category. Exercises. Historical Notes.
Part III. Selected Topics
27. Combinatorial Principles in I 545
The Fine Structure Theory. The Principle ?«. The Jensen Hierarchy. Projecta,
Standard Codes and Standard Parameters. Diamond Principles. Trees in L.
Canonical Functions on uii. Exercises. Historical Notes.
28. More Applications of Forcing 557
A Nonconstructible A3 Real. Namba Forcing. A Cohen Real Adds a Suslin
Tree. Consistency of Borel s Conjecture. K+ Aronszajn Trees. Exercises. His¬
torical Notes.
29. More Combinatorial Set Theory 573
Ramsey Theory. Gaps in u . The Open Coloring Axiom. Almost Disjoint
Subsets of wj. Functions from wi into w. Exercises. Historical Notes.
30. Complete Boolean Algebras 585
Measure Algebras. Cohen Algebras. Suslin Algebras. Simple Algebras. Infinite
Games on Boolean Algebras. Exercises. Historical Notes.
31. Proper Forcing 601
Definition and Examples. Iteration of Proper Forcing. The Proper Forcing
Axiom. Applications of PFA. Exercises. Historical Notes.
32. More Descriptive Set Theory 615
III Equivalence Relations. S| Equivalence Relations. Constructible Reals
and Perfect Sets. Projective Sets and Large Cardinals. Universally Baire sets.
Exercises. Historical Notes.
33. Determinacy 627
Determinacy and Choice. Some Consequences of AD. AD and Large Cardinals.
Projective Determinacy. Consistency of AD. Exercises. Historical Notes.
34. Supercompact Cardinals and the Real Line 647
Woodin Cardinals. Semiproper Forcing. The Model L(R). Stationary Tower
Forcing. Weakly Homogeneous Trees. Exercises. Historical Notes.
35. Inner Models for Large Cardinals 659
The Core Model. The Covering Theorem for K. The Covering Theorem
for L U . The Core Model for Sequences of Measures. Up to a Strong Cardinal.
Inner Models for Woodin Cardinals. Exercises. Historical Notes.
Table of Contents XIII
36. Forcing and Large Cardinals 669
Violating GCH at a Measurable Cardinal. The Singular Cardinal Problem.
Violating SCH at Ku,. Radin Forcing. Stationary Tower Forcing. Exercises.
Historical Notes.
37. Martin s Maximum 681
RCS iteration of semiproper forcing. Consistency of MM. Applications of MM.
Reflection Principles. Forcing Axioms. Exercises. Historical Notes.
38. More on Stationary Sets 695
The Nonstationary Ideal on Ni. Saturation and Precipitousness. Reflection.
Stationary Sets in PK(A). Mutually Stationary Sets. Weak Squares. Exercises.
Historical Notes.
Bibliography 707
Notation 733
Name Index 743
Index 749
|
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| author | Jech, Thomas J. 1944- |
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| classification_tum | MAT 040f |
| ctrlnum | (OCoLC)248757901 (DE-599)BVBBV014685961 |
| dewey-full | 511.322 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 511 - General principles of mathematics |
| dewey-raw | 511.322 |
| dewey-search | 511.322 |
| dewey-sort | 3511.322 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 3. millenium ed, rev. and expanded |
| format | Book |
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| id | DE-604.BV014685961 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T11:06:29Z |
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| language | English |
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| spellingShingle | Jech, Thomas J. 1944- Set theory Mengenlehre Set theory Axiomatische Mengenlehre (DE-588)4143743-3 gnd Mengenlehre (DE-588)4074715-3 gnd |
| subject_GND | (DE-588)4143743-3 (DE-588)4074715-3 |
| title | Set theory |
| title_auth | Set theory |
| title_exact_search | Set theory |
| title_full | Set theory Thomas Jech |
| title_fullStr | Set theory Thomas Jech |
| title_full_unstemmed | Set theory Thomas Jech |
| title_short | Set theory |
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| topic | Mengenlehre Set theory Axiomatische Mengenlehre (DE-588)4143743-3 gnd Mengenlehre (DE-588)4074715-3 gnd |
| topic_facet | Mengenlehre Set theory Axiomatische Mengenlehre |
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