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Handbook of Set Theory

E-BookPDF1 - PDF WatermarkE-Book
2230 Seiten
Englisch
Springer Netherlandserschienen am10.12.20092010
Numbers imitate space, which is of such a di?erent nature -Blaise Pascal It is fair to date the study of the foundation of mathematics back to the ancient Greeks. The urge to understand and systematize the mathematics of the time led Euclid to postulate axioms in an early attempt to put geometry on a ?rm footing. With roots in the Elements, the distinctive methodology of mathematics has become proof. Inevitably two questions arise: What are proofs? and What assumptions are proofs based on? The ?rst question, traditionally an internal question of the ?eld of logic, was also wrestled with in antiquity. Aristotle gave his famous syllogistic s- tems, and the Stoics had a nascent propositional logic. This study continued with ?ts and starts, through Boethius, the Arabs and the medieval logicians in Paris and London. The early germs of logic emerged in the context of philosophy and theology. The development of analytic geometry, as exempli?ed by Descartes, ill- tratedoneofthedi?cultiesinherentinfoundingmathematics. Itisclassically phrased as the question ofhow one reconciles the arithmetic with the geom- ric. Arenumbers onetypeofthingand geometricobjectsanother? Whatare the relationships between these two types of objects? How can they interact? Discovery of new types of mathematical objects, such as imaginary numbers and, much later, formal objects such as free groups and formal power series make the problem of ?nding a common playing ?eld for all of mathematics importunate. Several pressures made foundational issues urgent in the 19th century.mehr
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KlappentextNumbers imitate space, which is of such a di?erent nature -Blaise Pascal It is fair to date the study of the foundation of mathematics back to the ancient Greeks. The urge to understand and systematize the mathematics of the time led Euclid to postulate axioms in an early attempt to put geometry on a ?rm footing. With roots in the Elements, the distinctive methodology of mathematics has become proof. Inevitably two questions arise: What are proofs? and What assumptions are proofs based on? The ?rst question, traditionally an internal question of the ?eld of logic, was also wrestled with in antiquity. Aristotle gave his famous syllogistic s- tems, and the Stoics had a nascent propositional logic. This study continued with ?ts and starts, through Boethius, the Arabs and the medieval logicians in Paris and London. The early germs of logic emerged in the context of philosophy and theology. The development of analytic geometry, as exempli?ed by Descartes, ill- tratedoneofthedi?cultiesinherentinfoundingmathematics. Itisclassically phrased as the question ofhow one reconciles the arithmetic with the geom- ric. Arenumbers onetypeofthingand geometricobjectsanother? Whatare the relationships between these two types of objects? How can they interact? Discovery of new types of mathematical objects, such as imaginary numbers and, much later, formal objects such as free groups and formal power series make the problem of ?nding a common playing ?eld for all of mathematics importunate. Several pressures made foundational issues urgent in the 19th century.
Details
Weitere ISBN/GTIN9781402057649
ProduktartE-Book
EinbandartE-Book
FormatPDF
Format Hinweis1 - PDF Watermark
FormatE107
Erscheinungsjahr2009
Erscheinungsdatum10.12.2009
Auflage2010
Seiten2230 Seiten
SpracheEnglisch
IllustrationenXIV, 2230 p.
Artikel-Nr.1011520
Rubriken
Genre9200

Inhalt/Kritik

Inhaltsverzeichnis
1;Preface;5
2;Contents;11
3;List of Contributors;13
4;Introduction;15
4.1;Beginnings;16
4.1.1;Cantor;16
4.1.2;Zermelo;18
4.1.3;First Developments;20
4.1.4;Replacement and Foundation;24
4.2;New Groundwork;27
4.2.1;Gödel;27
4.2.2;Infinite Combinatorics;31
4.2.3;Definability;33
4.2.4;Model-Theoretic Techniques;36
4.3;The Advent of Forcing;41
4.3.1;Cohen;41
4.3.2;Method of Forcing;43
4.3.3;0#, L[U], and L[U];47
4.3.4;Constructibility;50
4.4;Strong Hypotheses;55
4.4.1;Large Large Cardinals;55
4.4.2;Determinacy;58
4.4.3;Silver's Theorem and Covering;62
4.4.4;Forcing Consistency Results;66
4.5;New Expansion;70
4.5.1;Into the 1980s;70
4.5.2;Consistency of Determinacy;74
4.5.3;Later Developments;77
4.6;Summaries of the Handbook Chapters;82
5;Stationary Sets;107
5.1;The Closed Unbounded Filter;108
5.1.1;Closed Unbounded Sets;108
5.1.2;Splitting Stationary Sets;110
5.1.3;Generic Ultrapowers;111
5.1.4;Stationary Sets in Generic Extensions;113
5.1.5;Some Combinatorial Principles;114
5.2;Reflection;114
5.2.1;Reflecting Stationary Sets;114
5.2.2;A Hierarchy of Stationary Sets;117
5.2.3;Canonical Stationary Sets;118
5.2.4;Full Reflection;119
5.3;Saturation;120
5.3.1;kappa+-saturation;120
5.3.2;Precipitousness;123
5.4;The Closed Unbounded Filter on Pkappalambda;124
5.4.1;Closed Unbounded Sets in PkappaA;124
5.4.2;Splitting Stationary Sets;127
5.4.3;Saturation;128
5.5;Proper Forcing and Other Applications;129
5.5.1;Proper Forcing;129
5.5.2;Projective and Cohen Boolean Algebras;130
5.6;Reflection;132
5.6.1;Reflection Principles;132
5.6.2;Nonreflecting Stationary Sets;135
5.7;Stationary Tower Forcing;135
5.8;Bibliography;136
6;Partition Relations;143
6.1;Introduction;144
6.1.1;Basic Definitions;146
6.2;Basic Partition Relations;148
6.2.1;Ramsey's Theorem;148
6.2.2;Ramification Arguments;149
6.2.3;Negative Stepping Up Lemma;151
6.3;Partition Relations and Submodels;151
6.4;Generalizations of the Erdos-Rado Theorem;154
6.4.1;Overview;154
6.4.2;More Elementary Submodels;157
6.4.3;The Balanced Generalization;158
6.4.4;The Unbalanced Generalization;161
6.4.5;The Baumgartner-Hajnal Theorem;166
6.5;The Milner-Rado Paradox and Omega(kappa);174
6.6;Shelah's Theorem for Infinitely Many Colors;176
6.7;Singular Cardinal Resources;179
6.8;Polarized Partition Relations;181
6.8.1;Successors of Weakly Compact Cardinals;181
6.8.2;Successors of Singular Cardinals;185
6.9;Countable Ordinal Resources;190
6.9.1;Some History;190
6.9.2;Small Counterexamples;191
6.10;A Positive Countable Partition Relation;201
6.10.1;Representation;202
6.10.2;Node Labeled Trees;206
6.10.3;Game;208
6.10.4;Uniformization;210
6.10.5;Triangles;213
6.10.6;Free Sets;218
6.10.7;Completion of the Proof;223
6.11;Bibliography;223
7;Coherent Sequences;228
7.1;The Space of Countable Ordinals;230
7.2;Subadditive Functions;237
7.3;Steps and Coherence;244
7.4;The Trace and the Square-Bracket Operation;249
7.5;A Square-Bracket Operation on a Tree;256
7.6;Special Trees and Mahlo Cardinals;257
7.7;The Weight Function on Successor Cardinals;261
7.8;The Number of Steps;263
7.9;Square Sequences;265
7.10;The Full Lower Trace of a Square Sequence;270
7.11;Special Square Sequences;273
7.12;Successors of Regular Cardinals;276
7.13;Successors of Singular Cardinals;280
7.14;The Oscillation Mapping;286
7.15;The Square-Bracket Operation;288
7.16;Unbounded Functions on Successors of Regular Cardinals;294
7.17;Higher Dimensions;299
7.18;Bibliography;305
8;Borel Equivalence Relations;310
8.1;Definitions;311
8.2;A Survey of Structure Theorems;319
8.2.1;Structure;319
8.2.2;Anti-Structure;323
8.2.3;Beyond Good and Evil;324
8.3;Countable Borel Equivalence Relations;326
8.3.1;The Global Structure;327
8.3.2;Treeable Equivalence Relations;334
8.3.3;Hyperfiniteness;335
8.4;Effective Cardinality;336
8.5;Classification Problems;337
8.5.1;Smooth versus Non-Smooth;338
8.5.2;Universal for Polish Group Actions;338
8.5.3;Universal for Sinfty;339
8.5.4;Einfty;340
8.5.5;E0;340
8.6;Bibliography;341
9;Proper Forcing;346
9.1;Introduction;347
9.1.1;Countable Support Iterations;352
9.2;Properness and Its Iteration;353
9.2.1;Preservation of Properness;357
9.2.2;The 2-Chain Condition;360
9.2.3;Equivalent Formulations;363
9.3;Preservation of omegaomega-Boundedness;365
9.3.1;Application: Non-Isomorphism of Ultrapowers;369
9.4;Preservation of Unboundedness;374
9.4.1;The Almost Bounding Property;376
9.4.2;Application to Cardinal Invariants;377
9.5;No New Reals;382
9.5.1;alpha-Properness;382
9.5.1.1;Equivalent Definition;383
9.5.1.2;Preservation of alpha-Properness;384
9.5.2;A Coloring Problem;387
9.5.3;Dee-Completeness;392
9.5.3.1;Two-Step Iteration;396
9.5.3.2;Proof of Theorem 5.17;398
9.5.3.3;Simple Completeness Systems;401
9.5.4;The Properness Isomorphism Condition;403
9.6;Bibliography;406
10;Combinatorial Cardinal Characteristics of the Continuum;408
10.1;Introduction;409
10.2;Growth of Functions;411
10.3;Splitting and Homogeneity;415
10.4;Galois-Tukey Connections and Duality;421
10.5;Category and Measure;430
10.6;Sparse Sets of Integers;438
10.7;Forcing Axioms;447
10.8;Almost Disjoint and Independent Families;455
10.9;Filters and Ultrafilters;459
10.10;Evasion and Prediction;471
10.11;Forcing;481
10.11.1;Finite Support Iteration and Martin's Axiom;483
10.11.2;Countable Support Proper Iteration;483
10.11.3;Cohen Reals;485
10.11.4;Random Reals;486
10.11.5;Sacks Reals;488
10.11.6;Hechler Reals;490
10.11.7;Laver Reals;491
10.11.8;Mathias Reals;491
10.11.9;Miller Reals;492
10.11.10;Summary of Iterated Forcing Results;493
10.11.11;Other Forcing Iterations;493
10.11.12;Adding One Real;495
10.12;Bibliography;496
11;Invariants of Measure and Category;503
11.1;Introduction;504
11.2;Tukey Connections;505
11.3;Inequalities Provable in ZFC;507
11.4;Combinatorial Characterizations;519
11.5;Cofinality of cov(J) and COV(J);530
11.6;Consistency Results and Counterexamples;547
11.7;Further Reading;560
11.8;Bibliography;564
12;Constructibility and Class Forcing;568
12.1;Three Problems of Solovay;570
12.2;Tameness;572
12.3;Examples;576
12.4;Relevance;580
12.5;The Coding Theorem;588
12.6;The Solovay Problems;599
12.7;Generic Saturation;602
12.8;Further Results;605
12.9;Some Open Problems;613
12.10;Bibliography;614
13;Fine Structure;616
13.1;Acceptable J-structures;617
13.2;The First Projectum;629
13.3;Downward Extension of Embeddings;632
13.4;Upward Extension of Embeddings;636
13.5;Iterated Projecta;642
13.6;Standard Parameters;646
13.7;Solidity Witnesses;649
13.8;Fine Ultrapowers;652
13.9;Applications to L;662
13.10;Bibliography;666
14;Sigma Fine Structure;668
14.1;Sigma Fine Structure;669
14.1.1;Variant Fine Structures;682
14.1.2;Sigma Ultrapowers;684
14.1.3;Pseudo-Ultrapowers;696
14.2;Global ;703
14.2.1;Defining Cnu;712
14.2.2;Variants and Generalities on ;725
14.2.3; in Fine-Structural Inner Models;731
14.3;Morasses;735
14.3.1;Construction of Gap-1 Morasses in L;737
14.3.2;Variants;741
14.4;Bibliography;744
15;Elementary Embeddings and Algebra;748
15.1;Iterations of an Elementary Embedding;749
15.1.1;Kunen's Bound and Axiom (I3);749
15.1.2;Operations on Elementary Embeddings;751
15.1.3;Iterations of an Elementary Embedding;753
15.1.4;Finite Quotients;756
15.1.5;The Laver-Steel Theorem;760
15.1.6;Counting the Critical Ordinals;762
15.2;The Word Problem for Self-Distributivity;767
15.2.1;Iterated Left Division in LD-systems;767
15.2.2;Using Elementary Embeddings;770
15.2.3;Avoiding Elementary Embeddings;771
15.3;Periods in the Laver Tables;773
15.3.1;Finite LD-Systems;773
15.3.2;Using Elementary Embeddings;776
15.3.3;Avoiding Elementary Embeddings;779
15.3.4;Not Avoiding Elementary Embeddings?;779
15.4;Bibliography;784
16;Iterated Forcing and Elementary Embeddings;786
16.1;Introduction;787
16.2;Elementary Embeddings;792
16.3;Ultrapowers and Extenders;795
16.4;Large Cardinal Axioms;798
16.5;Forcing;800
16.6;Some Forcing Posets;805
16.7;Iterated Forcing;809
16.8;Building Generic Objects;814
16.9;Lifting Elementary Embeddings;817
16.10;Generic Embeddings;820
16.11;Iteration with Easton Support;822
16.12;Master Conditions;825
16.13;A Technique of Magidor;830
16.14;Absorption;831
16.15;Transfer and Pullback;839
16.16;Small Large Cardinals;840
16.17;Precipitous Ideals I;842
16.17.1;A Precipitous Ideal on omega1;842
16.17.2;Iterated Club Shooting;844
16.17.2.1;Overview;845
16.17.2.2;Details;845
16.17.2.3;Precipitousness;848
16.18;Precipitous Ideals II;851
16.18.1;A Lower Bound;851
16.18.2;Precipitousness for NSomega2 | Cof(omega1);852
16.18.3;Outline of the Proof and Main Technical Issues;853
16.18.4;Namba Forcing, RCS Iteration and the S and I Conditions;855
16.18.5;The Preparation Iteration;857
16.18.6;A Warm-up for the Main Iteration;858
16.18.7;The Main Iteration;863
16.18.8;Precipitousness of the Non-Stationary Ideal;865
16.18.9;Successors of Larger Cardinals;866
16.19;More on Iterated Club Shooting;867
16.20;More on Collapses;871
16.21;Limiting Results;873
16.22;Termspace Forcing;876
16.23;More on Termspace Forcing and Collapsing;879
16.24;Iterations with Prediction;882
16.25;Altering Generic Objects;888
16.26;Bibliography;890
17;Ideals and Generic Elementary Embeddings;895
17.1;Introduction;897
17.1.1;An Overview of the Chapter;898
17.1.2;Apology and Acknowledgments;900
17.1.3;Some Conventions Used in the Chapter;901
17.2;Basic Facts;902
17.2.1;The Generic Ultrapower;904
17.2.2;Game Characterization of Precipitousness;908
17.2.3;Disjointing Property and Closure of Ultrapowers;908
17.2.4;Normal Ideals;910
17.2.4.1;Weak Normality;914
17.2.5;More General Facts;914
17.2.6;Canonical Functions;916
17.2.7;Selectivity;918
17.2.8;Ideals and Reflection;918
17.3;Examples;920
17.3.1;Natural Ideals;921
17.3.1.1;The Closed Unbounded Filter and the Nonstationary Ideal;921
17.3.1.2;Natural Ideals on P(R);927
17.3.1.3;I[lambda] and Related Ideals;928
17.3.1.4;Club Guessing Ideals;930
17.3.1.5;Ideals of Sets Without Guessing Sequences;931
17.3.1.6;Uniformization Ideals;933
17.3.1.7;Weakly Compact and Ineffable Ideals;934
17.3.2;Induced Ideals;934
17.3.2.1;General Induced Ideals;937
17.3.2.2;Goodness and Self-Genericity;938
17.4;A Closer Look;941
17.4.1;A Structural Property of Saturated Ideals;941
17.4.2;Saturation Properties;942
17.4.3;Layered Ideals;947
17.4.4;Projections;951
17.4.5;Where the Ordinals Go;955
17.4.6;A Discussion of Large Sets;958
17.4.7;Iterating Ideals;962
17.4.8;Generic Ultrapowers by Towers;964
17.5;Consequences of Generic Large Cardinals;968
17.5.1;Using Reflection;968
17.5.2;Chang's Conjectures, Jónsson Cardinals and Square;970
17.5.3;Ideals and GCH;972
17.5.3.1;Woodin's Theorem Showing CH Holds;973
17.5.3.2;Abe's Results on SCH;977
17.5.3.3;The Value of Theta;978
17.5.4;Stationary Set Reflection;981
17.5.5;Suslin and Kurepa Trees;983
17.5.6;Partition Properties;985
17.5.7;The Normal Moore Space Conjecture and Variants;989
17.5.8;Consequences in Descriptive Set Theory;990
17.5.9;Connections with Non-Regular Ultrafilters;991
17.5.9.1;Fully Non-Regular Ultrafilters on omega2;994
17.5.10;Graphs and Chromatic Numbers;995
17.5.11;The Nonstationary Ideal on omega1;996
17.6;Some Limitations;999
17.6.1;Soft Limitations;1000
17.6.2;The "Kunen Argument";1002
17.6.3;Saturated Ideals and Cofinalities;1004
17.6.4;Closed Unbounded Subsets of [kappa]omega;1008
17.6.5;Uniform Ideals on Ordinals;1009
17.6.6;Restrictions on the Quotient Algebra;1011
17.6.7;Yet Another Result of Kunen;1017
17.6.8;The Matsubara-Shioya Theorem;1018
17.6.9;The Nonstationary Ideal on [lambda]µ+;1332
19.4.8;Trees at Successors of Singular Cardinals;1335
19.5;Square-Bracket Partition Relations;1336
19.5.1;Colorings of Finite Subsets;1337
19.5.2;Colorings of Pairs;1341
19.5.3;Colorings and Club Guessing;1346
19.6;Concluding Remarks;1352
19.7;Bibliography;1353
20;Prikry-Type Forcings;1359
20.1;Prikry Forcings;1360
20.1.1;Basic Prikry Forcing;1360
20.1.2;Tree Prikry Forcing;1364
20.1.3;Adding a Prikry Sequence to a Singular Cardinal;1367
20.1.4;Supercompact and Strongly Compact Prikry Forcings;1369
20.2;Adding Many Prikry Sequences to a Singular Cardinal;1373
20.3;Extender-Based Prikry Forcing with a Single Extender;1386
20.4;Down to omega;1399
20.5;Forcing Uncountable Cofinalities;1407
20.5.1;Radin Forcing;1408
20.5.2;Magidor Forcing and Coherent Sequences of Measures;1425
20.5.3;Extender-Based Radin Forcing;1427
20.6;Iterations of Prikry-Type Forcing Notions;1432
20.6.1;Magidor Iteration;1432
20.6.2;Leaning's Forcing;1440
20.6.3;Easton Support Iteration;1441
20.6.3.1;Successor Levels;1446
20.6.3.2;Limit Levels;1446
20.6.4;An Application to Distributive Forcing Notions;1449
20.7;Some Open Problems;1451
20.8;Bibliography;1452
21;Beginning Inner Model Theory;1456
21.1;The Constructible Sets;1458
21.1.1;Relative Constructibility;1460
21.1.2;Measurable Cardinals;1461
21.1.3;0#, and Sharps in General;1465
21.1.4;Other Sharps;1468
21.1.5;From Sharps to the Core Model;1470
21.2;Beyond One Measurable Cardinal;1471
21.2.1;The Comparison Process;1473
21.2.2;Indiscernibles from Iterated Ultrapowers;1478
21.3;Extender Models;1480
21.3.1;The Modern Presentation of L[E];1491
21.4;Remarks on Larger Cardinals;1493
21.4.1;Strong cardinals;1494
21.4.2;Woodin cardinals;1494
21.4.3;Superstrong Cardinals;1495
21.4.4;Supercompact Cardinals;1495
21.4.5;Larger Cardinals;1496
21.5;What is the Core Model?;1496
21.6;Bibliography;1500
22;The Covering Lemma;1503
22.1;The Statement;1504
22.1.1;The Weak Covering Lemma;1507
22.1.2;The Strong Covering Lemma;1508
22.1.3;The Covering Lemma without Second-Order Closure;1509
22.2;Basic Applications;1510
22.2.1;The Weak Covering Lemma;1511
22.2.2;The Full Covering Lemma;1512
22.3;The Proof;1514
22.3.1;Fine Structure and Other Tools;1515
22.3.1.1;Embeddings of Mice;1521
22.3.2;Proof of the Covering Lemma for L;1523
22.3.2.1;Suitable Sets;1525
22.3.3;Measurable Cardinals;1529
22.3.3.1;Comparisons of Mice;1539
22.3.3.2;The Dodd-Jensen Core Model;1544
22.3.3.3;Part 1 of the Proof;1548
22.3.3.4;Part 2 of the Proof: Analyzing the Indiscernibles;1553
22.3.4;Unsuitable Covering Sets;1557
22.4;Sequences of Measures;1560
22.4.1;The Covering Lemma and Sequences of Measures;1560
22.4.2;Extenders;1563
22.4.3;The Core Model for Sequences of Measures;1564
22.4.4;The Covering Lemma up to o(kappa)=kappa++;1570
22.4.4.1;Introduction to the Proof;1573
22.4.4.2;Part 1 of the Proof;1574
22.4.4.3;Part 2 of the Proof: Analyzing the Indiscernibles;1576
22.4.4.4;Digression for Non-Countably Closed Sets X;1578
22.4.4.5;Continuation of the Main Proof;1580
22.4.5;The Singular Cardinal Hypothesis;1585
22.4.6;The Covering Lemma for Extenders;1590
22.4.6.1;Up to a Strong Cardinal;1590
22.4.6.2;Up to a Woodin Cardinal;1594
22.4.6.3;Beyond a Woodin Cardinal;1596
22.5;Bibliography;1596
23;An Outline of Inner Model Theory;1601
23.1;Introduction;1602
23.2;Premice;1602
23.2.1;Extenders;1603
23.2.2;Fine Extender Sequences;1605
23.2.3;The Levy Hierarchy, Cores, and Soundness;1609
23.2.4;Fine Structure and Ultrapowers ;1615
23.3;Iteration Trees and Comparison;1617
23.3.1;Iteration Trees;1618
23.3.2;The Comparison Process ;1623
23.4;The Dodd-Jensen Lemma;1628
23.4.1;The Copying Construction;1628
23.4.2;The Dodd-Jensen Lemma ;1632
23.4.3;The Weak Dodd-Jensen Property;1634
23.5;Solidity and Condensation;1637
23.6;Background-Certified Fine Extender Sequences;1644
23.6.1;Kc-Constructions;1644
23.6.2;The Iterability of Kc;1647
23.6.3;Large Cardinals in Kc;1655
23.7;The Reals of Momega;1657
23.7.1;Iteration Strategies in L(R);1658
23.7.2;Correctness and Genericity Iterations;1663
23.8;HODL(R) below Theta;1674
23.9;Bibliography;1688
24;A Core Model Toolbox and Guide;1691
24.1;Introduction;1692
24.2;Basic Theory of K;1694
24.2.1;Second-Order Definition of K;1694
24.2.2;First-Order Definition of K;1713
24.3;Core Model Tools;1722
24.3.1;Covering Properties;1722
24.3.2;Absoluteness, Complexity and Correctness;1723
24.3.3;Embeddings of K;1724
24.3.4;Maximality;1725
24.3.5;Combinatorial Principles;1725
24.3.6;On the Technical Hypothesis;1726
24.4;Proof of Weak Covering;1727
24.5;Applications of Core Models;1743
24.5.1;Determinacy;1743
24.5.2;Tree Representations and Absoluteness;1746
24.5.3;Ideals and Generic Embeddings;1748
24.5.4;Square and Aronszajn Trees;1749
24.5.5;Forcing Axioms;1751
24.5.6;The Failure of UBH;1753
24.5.7;Cardinality and Cofinality;1754
24.6;Bibliography;1754
25;Structural Consequences of AD;1758
25.1;Introduction;1759
25.2;Survey of Basic Notions;1765
25.2.1;Prewellordering, Scales, and Periodicity;1765
25.2.2;Projective Ordinals, Sets, and the Coding Lemma;1774
25.2.3;Wadge Degrees and Abstract Pointclasses;1778
25.2.4;The Scale Theory of L(R);1783
25.2.5;Determinacy and Coding Results;1784
25.2.6;Partition Relations;1787
25.3;Suslin Cardinals;1790
25.3.1;Pointclass Arguments;1791
25.3.2;The Next Suslin Cardinal;1798
25.3.3;More on Lambda in the Type IV Case;1806
25.3.4;The Classification of the Suslin Cardinals;1809
25.4;Trivial Descriptions: A Theory of omega1;1811
25.4.1;Analysis of Measures on delta11;1815
25.4.2;The Strong Partition Relation on omega1;1818
25.4.3;The Weak Partition Relation on delta13;1822
25.4.4;The Kechris-Martin Theorem Revisited;1834
25.5;Higher Descriptions;1841
25.5.1;Martin's Theorem on Normal Measures;1841
25.5.2;Some Canonical Measures;1852
25.5.3;The Higher Descriptions;1854
25.5.4;Some Further Results;1865
25.6;Global Results;1866
25.6.1;Generic Codes;1867
25.6.2;Weak Square and Uniform Cofinalities;1871
25.6.3;Some Final Remarks;1876
25.7;Bibliography;1878
26;Determinacy in L(R);1882
26.1;Extenders and Iteration Trees;1886
26.2;Iterability;1894
26.3;Creating Iteration Trees;1901
26.4;Homogeneously Suslin Sets;1908
26.5;Projections and Complementations;1914
26.6;Universally Baire Sets;1926
26.7;Genericity Iterations;1933
26.8;Determinacy in L(R);1945
26.9;Bibliography;1953
27;Large Cardinals from Determinacy;1956
27.1;Introduction;1957
27.1.1;Determinacy and Large Cardinals;1957
27.1.1.1;A. Determinacy;1957
27.1.1.2;B. Large Cardinals;1960
27.1.1.3;C. Determinacy from Large Cardinals;1962
27.1.1.4;D. Large Cardinals from Determinacy;1963
27.1.1.5;E. Overview;1966
27.1.1.6;Acknowledgments;1968
27.1.2;Notation;1968
27.2;Basic Results;1970
27.2.1;Preliminaries;1970
27.2.2;Boundedness and Basic Coding;1973
27.2.3;Measurability;1977
27.2.4;The Least Stable;1982
27.2.5;Measurability of the Least Stable;1989
27.3;Coding;1992
27.3.1;Coding Lemma;1992
27.3.2;Uniform Coding Lemma;1996
27.3.3;Applications;2001
27.4;A Woodin Cardinal in HODL(R);2004
27.4.1;Reflection;2005
27.4.2;Strong Normality;2016
27.4.3;A Woodin Cardinal;2032
27.5;Woodin Cardinals in General Settings;2036
27.5.1;First Abstraction;2038
27.5.2;Strategic Determinacy;2040
27.5.3;Generation Theorem;2048
27.5.4;Special Cases;2072
27.6;Definable Determinacy;2079
27.6.1;Lightface Definable Determinacy;2080
27.6.2;Boldface Definable Determinacy;2099
27.7;Second-Order Arithmetic;2106
27.7.1;First Localization;2107
27.7.2;Second Localization;2113
27.8;Further Results;2114
27.8.1;Large Cardinals and Determinacy;2114
27.8.2;HOD-Analysis;2118
27.9;Bibliography;2124
28;Forcing over Models of Determinacy;2125
28.1;Iterations;2127
28.2;Pmax;2134
28.3;Sequences of Models and Countable Closure;2138
28.4;Generalized Iterability;2141
28.5;The Basic Analysis;2149
28.6;psiAC and the Axiom of Choice;2155
28.7;Maximality and Minimality;2156
28.8;Larger Models;2163
28.9;Omega-Logic;2166
28.10;Variations;2171
28.10.1;Variations for NSomega1;2171
28.10.2;Conditional Variations for Sigma2 sentences;2175
28.11;Bibliography;2178mehr
Leseprobe
Introduction (p. 1-2)

field of mathematics with broad foundational significance, and this Handbook with its expanse and variety amply attests to the fecundity and sophistication of the subject. Indeed, in set theory´s further reaches one sees tremendous progress both in its continuing development of its historical heritage, the investigation of the transfinite numbers and of definable sets of reals, as well as its analysis of strong propositions and consistency strength in terms of large cardinal hypotheses and inner models.

This introduction provides a historical and organizational frame for both modern set theory and this Handbook, the chapter summaries at the end being a final elaboration. To the purpose of drawing in the serious, mathematically experienced reader and providing context for the prospective researcher, we initially recapitulate the consequential historical developments leading to modern set theory as a field of mathematics. In the process we affirm basic concepts and terminology, chart out the motivating issues and driving initiatives, and describe the salient features of the field´s internal practices. As the narrative proceeds, there will be a natural inversion: Less and less will be said about more and more as one progresses from basic concepts to elaborate structures, from seminal proofs to complex argumentation, from individual moves to collective enterprise. We try to put matters in a succinct yet illuminating manner, but be that as it may, according to one´s experience or interest one can skim the all too familiar or too obscure. To the historian this account would not properly be history-t is, rather, a deliberate arrangement, in significant part to lay the ground for the coming chapters.

To the seasoned set theorist there may be issues of under-emphasis or overemphasis, of omissions or commissions. In any case, we take refuge in a wise aphorism: If it´s worth doing, it´s worth doing badly.

1. Beginnings

1.1. Cantor

Set theory was born on that day in December 1873 when Georg Cantor (1845-918) established that the continuum is not countable-here is no one-to-one correspondence between the real numbers and the natural numbers 0, 1, 2, . . . . Given a (countable) sequence of reals, Cantor defined nested intervals so that any real in their intersection will not be in the sequence. In the course of his earlier investigations of trigonometric series Cantor had developed a definition of the reals and had begun to entertain infinite totalities of reals for their own sake. Now with his uncountability result Cantor embarked on a full-fledged investigation that would initiate an expansion of the very concept of number. Articulating cardinality as based on bijection (one-to-one correspondence) Cantor soon established positive results about the existence of bijections between sets of reals, subsets of the plane, and the like. By 1878 his investigations had led him to assert that there are only twomehr