Handbook of Set Theory (eBook)

Preface 5
Contents 11
List of Contributors 13
Introduction 15
Beginnings 16
Cantor 16
Zermelo 18
First Developments 20
Replacement and Foundation 24
New Groundwork 27
Gödel 27
Infinite Combinatorics 31
Definability 33
Model-Theoretic Techniques 36
The Advent of Forcing 41
Cohen 41
Method of Forcing 43
0#, L[U], and L[U] 47
Constructibility 50
Strong Hypotheses 55
Large Large Cardinals 55
Determinacy 58
Silver's Theorem and Covering 62
Forcing Consistency Results 66
New Expansion 70
Into the 1980s 70
Consistency of Determinacy 74
Later Developments 77
Summaries of the Handbook Chapters 82
Stationary Sets 107
The Closed Unbounded Filter 108
Closed Unbounded Sets 108
Splitting Stationary Sets 110
Generic Ultrapowers 111
Stationary Sets in Generic Extensions 113
Some Combinatorial Principles 114
Reflection 114
Reflecting Stationary Sets 114
A Hierarchy of Stationary Sets 117
Canonical Stationary Sets 118
Full Reflection 119
Saturation 120
kappa+-saturation 120
Precipitousness 123
The Closed Unbounded Filter on Pkappalambda 124
Closed Unbounded Sets in PkappaA 124
Splitting Stationary Sets 127
Saturation 128
Proper Forcing and Other Applications 129
Proper Forcing 129
Projective and Cohen Boolean Algebras 130
Reflection 132
Reflection Principles 132
Nonreflecting Stationary Sets 135
Stationary Tower Forcing 135
Bibliography 136
Partition Relations 143
Introduction 144
Basic Definitions 146
Basic Partition Relations 148
Ramsey's Theorem 148
Ramification Arguments 149
Negative Stepping Up Lemma 151
Partition Relations and Submodels 151
Generalizations of the Erdos-Rado Theorem 154
Overview 154
More Elementary Submodels 157
The Balanced Generalization 158
The Unbalanced Generalization 161
The Baumgartner-Hajnal Theorem 166
The Milner-Rado Paradox and Omega(kappa) 174
Shelah's Theorem for Infinitely Many Colors 176
Singular Cardinal Resources 179
Polarized Partition Relations 181
Successors of Weakly Compact Cardinals 181
Successors of Singular Cardinals 185
Countable Ordinal Resources 190
Some History 190
Small Counterexamples 191
A Positive Countable Partition Relation 201
Representation 202
Node Labeled Trees 206
Game 208
Uniformization 210
Triangles 213
Free Sets 218
Completion of the Proof 223
Bibliography 223
Coherent Sequences 228
The Space of Countable Ordinals 230
Subadditive Functions 237
Steps and Coherence 244
The Trace and the Square-Bracket Operation 249
A Square-Bracket Operation on a Tree 256
Special Trees and Mahlo Cardinals 257
The Weight Function on Successor Cardinals 261
The Number of Steps 263
Square Sequences 265
The Full Lower Trace of a Square Sequence 270
Special Square Sequences 273
Successors of Regular Cardinals 276
Successors of Singular Cardinals 280
The Oscillation Mapping 286
The Square-Bracket Operation 288
Unbounded Functions on Successors of Regular Cardinals 294
Higher Dimensions 299
Bibliography 305
Borel Equivalence Relations 310
Definitions 311
A Survey of Structure Theorems 319
Structure 319
Anti-Structure 323
Beyond Good and Evil 324
Countable Borel Equivalence Relations 326
The Global Structure 327
Treeable Equivalence Relations 334
Hyperfiniteness 335
Effective Cardinality 336
Classification Problems 337
Smooth versus Non-Smooth 338
Universal for Polish Group Actions 338
Universal for Sinfty 339
Einfty 340
E0 340
Bibliography 341
Proper Forcing 346
Introduction 347
Countable Support Iterations 352
Properness and Its Iteration 353
Preservation of Properness 357
The 2-Chain Condition 360
Equivalent Formulations 363
Preservation of omegaomega-Boundedness 365
Application: Non-Isomorphism of Ultrapowers 369
Preservation of Unboundedness 374
The Almost Bounding Property 376
Application to Cardinal Invariants 377
No New Reals 382
alpha-Properness 382
Equivalent Definition 383
Preservation of alpha-Properness 384
A Coloring Problem 387
Dee-Completeness 392
Two-Step Iteration 396
Proof of Theorem 5.17 398
Simple Completeness Systems 401
The Properness Isomorphism Condition 403
Bibliography 406
Combinatorial Cardinal Characteristics of the Continuum 408
Introduction 409
Growth of Functions 411
Splitting and Homogeneity 415
Galois-Tukey Connections and Duality 421
Category and Measure 430
Sparse Sets of Integers 438
Forcing Axioms 447
Almost Disjoint and Independent Families 455
Filters and Ultrafilters 459
Evasion and Prediction 471
Forcing 481
Finite Support Iteration and Martin's Axiom 483
Countable Support Proper Iteration 483
Cohen Reals 485
Random Reals 486
Sacks Reals 488
Hechler Reals 490
Laver Reals 491
Mathias Reals 491
Miller Reals 492
Summary of Iterated Forcing Results 493
Other Forcing Iterations 493
Adding One Real 495
Bibliography 496
Invariants of Measure and Category 503
Introduction 504
Tukey Connections 505
Inequalities Provable in ZFC 507
Combinatorial Characterizations 519
Cofinality of cov(J) and COV(J) 530
Consistency Results and Counterexamples 547
Further Reading 560
Bibliography 564
Constructibility and Class Forcing 568
Three Problems of Solovay 570
Tameness 572
Examples 576
Relevance 580
The Coding Theorem 588
The Solovay Problems 599
Generic Saturation 602
Further Results 605
Some Open Problems 613
Bibliography 614
Fine Structure 616
Acceptable J-structures 617
The First Projectum 629
Downward Extension of Embeddings 632
Upward Extension of Embeddings 636
Iterated Projecta 642
Standard Parameters 646
Solidity Witnesses 649
Fine Ultrapowers 652
Applications to L 662
Bibliography 666
Sigma Fine Structure 668
Sigma Fine Structure 669
Variant Fine Structures 682
Sigma Ultrapowers 684
Pseudo-Ultrapowers 696
Global 703
Defining Cnu 712
Variants and Generalities on 725
in Fine-Structural Inner Models 731
Morasses 735
Construction of Gap-1 Morasses in L 737
Variants 741
Bibliography 744
Elementary Embeddings and Algebra 748
Iterations of an Elementary Embedding 749
Kunen's Bound and Axiom (I3) 749
Operations on Elementary Embeddings 751
Iterations of an Elementary Embedding 753
Finite Quotients 756
The Laver-Steel Theorem 760
Counting the Critical Ordinals 762
The Word Problem for Self-Distributivity 767
Iterated Left Division in LD-systems 767
Using Elementary Embeddings 770
Avoiding Elementary Embeddings 771
Periods in the Laver Tables 773
Finite LD-Systems 773
Using Elementary Embeddings 776
Avoiding Elementary Embeddings 779
Not Avoiding Elementary Embeddings? 779
Bibliography 784
Iterated Forcing and Elementary Embeddings 786
Introduction 787
Elementary Embeddings 792
Ultrapowers and Extenders 795
Large Cardinal Axioms 798
Forcing 800
Some Forcing Posets 805
Iterated Forcing 809
Building Generic Objects 814
Lifting Elementary Embeddings 817
Generic Embeddings 820
Iteration with Easton Support 822
Master Conditions 825
A Technique of Magidor 830
Absorption 831
Transfer and Pullback 839
Small Large Cardinals 840
Precipitous Ideals I 842
A Precipitous Ideal on omega1 842
Iterated Club Shooting 844
Overview 845
Details 845
Precipitousness 848
Precipitous Ideals II 851
A Lower Bound 851
Precipitousness for NSomega2 | Cof(omega1) 852
Outline of the Proof and Main Technical Issues 853
Namba Forcing, RCS Iteration and the S and I Conditions 855
The Preparation Iteration 857
A Warm-up for the Main Iteration 858
The Main Iteration 863
Precipitousness of the Non-Stationary Ideal 865
Successors of Larger Cardinals 866
More on Iterated Club Shooting 867
More on Collapses 871
Limiting Results 873
Termspace Forcing 876
More on Termspace Forcing and Collapsing 879
Iterations with Prediction 882
Altering Generic Objects 888
Bibliography 890
Ideals and Generic Elementary Embeddings 895
Introduction 897
An Overview of the Chapter 898
Apology and Acknowledgments 900
Some Conventions Used in the Chapter 901
Basic Facts 902
The Generic Ultrapower 904
Game Characterization of Precipitousness 908
Disjointing Property and Closure of Ultrapowers 908
Normal Ideals 910
Weak Normality 914
More General Facts 914
Canonical Functions 916
Selectivity 918
Ideals and Reflection 918
Examples 920
Natural Ideals 921
The Closed Unbounded Filter and the Nonstationary Ideal 921
Natural Ideals on P(R) 927
I[lambda] and Related Ideals 928
Club Guessing Ideals 930
Ideals of Sets Without Guessing Sequences 931
Uniformization Ideals 933
Weakly Compact and Ineffable Ideals 934
Induced Ideals 934
General Induced Ideals 937
Goodness and Self-Genericity 938
A Closer Look 941
A Structural Property of Saturated Ideals 941
Saturation Properties 942
Layered Ideals 947
Projections 951
Where the Ordinals Go 955
A Discussion of Large Sets 958
Iterating Ideals 962
Generic Ultrapowers by Towers 964
Consequences of Generic Large Cardinals 968
Using Reflection 968
Chang's Conjectures, Jónsson Cardinals and Square 970
Ideals and GCH 972
Woodin's Theorem Showing CH Holds 973
Abe's Results on SCH 977
The Value of Theta 978
Stationary Set Reflection 981
Suslin and Kurepa Trees 983
Partition Properties 985
The Normal Moore Space Conjecture and Variants 989
Consequences in Descriptive Set Theory 990
Connections with Non-Regular Ultrafilters 991
Fully Non-Regular Ultrafilters on omega2 994
Graphs and Chromatic Numbers 995
The Nonstationary Ideal on omega1 996
Some Limitations 999
Soft Limitations 1000
The "Kunen Argument" 1002
Saturated Ideals and Cofinalities 1004
Closed Unbounded Subsets of [kappa]omega 1008
Uniform Ideals on Ordinals 1009
Restrictions on the Quotient Algebra 1011
Yet Another Result of Kunen 1017
The Matsubara-Shioya Theorem 1018
The Nonstationary Ideal on [lambda]< kappa
Consistency Results 1024
Trivial Master Conditions 1024
The Basic Idea 1025
Precipitous Ideals on Accessible Cardinals 1027
Strong Master Conditions 1028
Precipitousness is not Preserved Under Projections 1029
Computing Quotient Algebras and Preserving Strong Ideals under Generic Extensions 1030
Preservation of Normality and a Warm-Up Theorem 1031
When Master Conditions Exist 1033
The Duality Theorem and Its Consequences 1034
Understanding the Embeddings: Master Conditions Must Exist 1037
Generalizations of the Duality Theorem Without Master Conditions 1039
Pseudo-Generic Towers 1039
A kappa-Saturated Ideal on an Inaccessible Cardinal kappa 1040
Basic Kunen Technique: kappa+-Saturated Ideals 1042
(2,2,0)-Saturated Ideals 1045
Chang Ideals with Simple Quotients I 1047
3-Dense Ideals on omega3 1048
Higher Chang's Conjectures and omega Jónsson 1049
The Magidor Variation 1050
More Saturated Ideals 1052
Forbidden Intervals 1059
Dense Ideals on omega1 1060
The Lower End 1062
Chang-Type Ideals with Simple Quotients II 1067
Destroying Precipitous and Saturated Ideals 1068
Consistency Results for Natural Ideals 1069
Forcing over Determinacy Models 1069
Making Induced Ideals Natural 1070
The Null and Meager Ideals 1071
Nonstationary Ideals 1072
Club Guessing Ideals 1076
Making Natural Ideals Have Well-Founded Ultrapowers 1081
Catching Antichains Using Reflection 1083
Reflecting Stationary Sets 1085
Catching Your Tail 1088
The Nonstationary Ideal is Precipitous 1089
Saturation of the Nonstationary Ideal 1090
The Equivalence of "Semiproper" with "Stationary Set Preserving" in the Case of Antichain Sealing Forcing 1092
Martin's Maximum and Related Topics 1095
Shelah's Results on Ulam's Problem 1097
Saturated Ideals and Square 1098
Tower Forcing 1099
Induced Towers 1101
General Techniques 1103
Good Structures 1103
Catching Antichains 1104
Catching Your Tail 1105
Using Antichain Catching 1106
Natural Towers 1109
Methods for Proving Antichain Capturing 1109
Woodin's Towers 1111
Burke's Towers 1114
Self-Genericity for Towers 1117
Examples of Stationary Tower Forcing 1119
A Tower that is not Precipitous 1124
Consistency Strength of Ideal Assumptions 1126
Fine Structural Inner Models 1126
Getting Very Large Cardinals from Ideal Hypotheses 1128
Constructing from Stationary Sets and the Nonstationary Ideal 1128
Decisive Ideals 1129
Chang's Conjecture and Huge Cardinals 1131
A Martin's Maximum Result 1133
Consistency Hierarchies Among Ideals 1134
Ideals as Axioms 1134
Generalized Large Cardinals 1135
Flies in the Ointment 1136
First-Order Statements in the Language of ZFC 1138
Some Examples of Axioms 1138
Coherence of Theories, Hierarchies of Strength and Predictions 1140
Predictions 1140
Gradations of Consequence 1142
Methodological Predictions 1143
A Final Relevant Issue 1143
Conclusion 1144
Open Questions 1144
Bibliography 1150
Cardinal Arithmetic 1158
Introduction 1159
Elementary Definitions 1160
Products of Sets 1162
Partial Orderings 1162
Projections 1165
Existence of Exact Upper Bounds 1167
Application: Silver's Theorem 1179
Application: A Covering Theorem 1180
Basic Properties of the pcf Function 1182
The Ideal J< lambda
Generators for J< lambda
The Cofinality of [µ]kappa 1196
Elementary Substructures 1197
Minimally Obedient Sequences 1200
Application: The Cofinality of ([µ]kappa, ) 1205
Elevations and Transitive Generators 1211
Localization 1214
Size Limitation on pcf of Intervals 1217
Revised GCH 1219
TD(f) 1225
Proof of the Revised GCH 1228
Applications of the Revised GCH 1229
Bibliography 1235
Successors of Singular Cardinals 1237
Introduction 1238
Three Problems 1238
The Cofinality of [omegaomega+1]0 1238
Reflection of Stationary Sets 1238
Is omega+1 a Jónsson cardinal? 1239
Conventions and Notation 1241
Elementary Submodels 1241
lambda-Filtration Sequences 1243
pcf Theory 1243
Large Cardinals 1247
Forcing 1248
Remarks 1248
On Stationary Reflection 1249
Squares and Supercompact Cardinals 1250
Reflection and Indecomposable Ultrafilters 1253
Reflection at omega+1-Introduction 1256
kappa+-closed Forcing and Stationary Subsets of Slambdakappa 1257
Reflection at omega+1-Conclusion 1261
On I[lambda] 1263
The Ideal I[lambda] 1264
Construction 1269
The Approachability Property 1270
The Extent of I[lambda] 1274
Weak Approachability and APomega 1277
The Structure of I[lambda] 1282
An Application-the Existence of Scales 1288
Applications of Scales and Weak Squares 1295
Weakenings of -Part I 1295
Weakenings of -Part II 1298
On 1304
Very Weak Square 1309
NPT and Good Scales 1314
Varieties of Nice Scales 1325
Some Consequences of pp(µ)> µ+
Trees at Successors of Singular Cardinals 1335
Square-Bracket Partition Relations 1336
Colorings of Finite Subsets 1337
Colorings of Pairs 1341
Colorings and Club Guessing 1346
Concluding Remarks 1352
Bibliography 1353
Prikry-Type Forcings 1359
Prikry Forcings 1360
Basic Prikry Forcing 1360
Tree Prikry Forcing 1364
Adding a Prikry Sequence to a Singular Cardinal 1367
Supercompact and Strongly Compact Prikry Forcings 1369
Adding Many Prikry Sequences to a Singular Cardinal 1373
Extender-Based Prikry Forcing with a Single Extender 1386
Down to omega 1399
Forcing Uncountable Cofinalities 1407
Radin Forcing 1408
Magidor Forcing and Coherent Sequences of Measures 1425
Extender-Based Radin Forcing 1427
Iterations of Prikry-Type Forcing Notions 1432
Magidor Iteration 1432
Leaning's Forcing 1440
Easton Support Iteration 1441
Successor Levels 1446
Limit Levels 1446
An Application to Distributive Forcing Notions 1449
Some Open Problems 1451
Bibliography 1452
Beginning Inner Model Theory 1456
The Constructible Sets 1458
Relative Constructibility 1460
Measurable Cardinals 1461
0#, and Sharps in General 1465
Other Sharps 1468
From Sharps to the Core Model 1470
Beyond One Measurable Cardinal 1471
The Comparison Process 1473
Indiscernibles from Iterated Ultrapowers 1478
Extender Models 1480
The Modern Presentation of L[E] 1491
Remarks on Larger Cardinals 1493
Strong cardinals 1494
Woodin cardinals 1494
Superstrong Cardinals 1495
Supercompact Cardinals 1495
Larger Cardinals 1496
What is the Core Model? 1496
Bibliography 1500
The Covering Lemma 1503
The Statement 1504
The Weak Covering Lemma 1507
The Strong Covering Lemma 1508
The Covering Lemma without Second-Order Closure 1509
Basic Applications 1510
The Weak Covering Lemma 1511
The Full Covering Lemma 1512
The Proof 1514
Fine Structure and Other Tools 1515
Embeddings of Mice 1521
Proof of the Covering Lemma for L 1523
Suitable Sets 1525
Measurable Cardinals 1529
Comparisons of Mice 1539
The Dodd-Jensen Core Model 1544
Part 1 of the Proof 1548
Part 2 of the Proof: Analyzing the Indiscernibles 1553
Unsuitable Covering Sets 1557
Sequences of Measures 1560
The Covering Lemma and Sequences of Measures 1560
Extenders 1563
The Core Model for Sequences of Measures 1564
The Covering Lemma up to o(kappa)=kappa++ 1570
Introduction to the Proof 1573
Part 1 of the Proof 1574
Part 2 of the Proof: Analyzing the Indiscernibles 1576
Digression for Non-Countably Closed Sets X 1578
Continuation of the Main Proof 1580
The Singular Cardinal Hypothesis 1585
The Covering Lemma for Extenders 1590
Up to a Strong Cardinal 1590
Up to a Woodin Cardinal 1594
Beyond a Woodin Cardinal 1596
Bibliography 1596
An Outline of Inner Model Theory 1601
Introduction 1602
Premice 1602
Extenders 1603
Fine Extender Sequences 1605
The Levy Hierarchy, Cores, and Soundness 1609
Fine Structure and Ultrapowers 1615
Iteration Trees and Comparison 1617
Iteration Trees 1618
The Comparison Process 1623
The Dodd-Jensen Lemma 1628
The Copying Construction 1628
The Dodd-Jensen Lemma 1632
The Weak Dodd-Jensen Property 1634
Solidity and Condensation 1637
Background-Certified Fine Extender Sequences 1644
Kc-Constructions 1644
The Iterability of Kc 1647
Large Cardinals in Kc 1655
The Reals of Momega 1657
Iteration Strategies in L(R) 1658
Correctness and Genericity Iterations 1663
HODL(R) below Theta 1674
Bibliography 1688
A Core Model Toolbox and Guide 1691
Introduction 1692
Basic Theory of K 1694
Second-Order Definition of K 1694
First-Order Definition of K 1713
Core Model Tools 1722
Covering Properties 1722
Absoluteness, Complexity and Correctness 1723
Embeddings of K 1724
Maximality 1725
Combinatorial Principles 1725
On the Technical Hypothesis 1726
Proof of Weak Covering 1727
Applications of Core Models 1743
Determinacy 1743
Tree Representations and Absoluteness 1746
Ideals and Generic Embeddings 1748
Square and Aronszajn Trees 1749
Forcing Axioms 1751
The Failure of UBH 1753
Cardinality and Cofinality 1754
Bibliography 1754
Structural Consequences of AD 1758
Introduction 1759
Survey of Basic Notions 1765
Prewellordering, Scales, and Periodicity 1765
Projective Ordinals, Sets, and the Coding Lemma 1774
Wadge Degrees and Abstract Pointclasses 1778
The Scale Theory of L(R) 1783
Determinacy and Coding Results 1784
Partition Relations 1787
Suslin Cardinals 1790
Pointclass Arguments 1791
The Next Suslin Cardinal 1798
More on Lambda in the Type IV Case 1806
The Classification of the Suslin Cardinals 1809
Trivial Descriptions: A Theory of omega1 1811
Analysis of Measures on delta11 1815
The Strong Partition Relation on omega1 1818
The Weak Partition Relation on delta13 1822
The Kechris-Martin Theorem Revisited 1834
Higher Descriptions 1841
Martin's Theorem on Normal Measures 1841
Some Canonical Measures 1852
The Higher Descriptions 1854
Some Further Results 1865
Global Results 1866
Generic Codes 1867
Weak Square and Uniform Cofinalities 1871
Some Final Remarks 1876
Bibliography 1878
Determinacy in L(R) 1882
Extenders and Iteration Trees 1886
Iterability 1894
Creating Iteration Trees 1901
Homogeneously Suslin Sets 1908
Projections and Complementations 1914
Universally Baire Sets 1926
Genericity Iterations 1933
Determinacy in L(R) 1945
Bibliography 1953
Large Cardinals from Determinacy 1956
Introduction 1957
Determinacy and Large Cardinals 1957
A. Determinacy 1957
B. Large Cardinals 1960
C. Determinacy from Large Cardinals 1962
D. Large Cardinals from Determinacy 1963
E. Overview 1966
Acknowledgments 1968
Notation 1968
Basic Results 1970
Preliminaries 1970
Boundedness and Basic Coding 1973
Measurability 1977
The Least Stable 1982
Measurability of the Least Stable 1989
Coding 1992
Coding Lemma 1992
Uniform Coding Lemma 1996
Applications 2001
A Woodin Cardinal in HODL(R) 2004
Reflection 2005
Strong Normality 2016
A Woodin Cardinal 2032
Woodin Cardinals in General Settings 2036
First Abstraction 2038
Strategic Determinacy 2040
Generation Theorem 2048
Special Cases 2072
Definable Determinacy 2079
Lightface Definable Determinacy 2080
Boldface Definable Determinacy 2099
Second-Order Arithmetic 2106
First Localization 2107
Second Localization 2113
Further Results 2114
Large Cardinals and Determinacy 2114
HOD-Analysis 2118
Bibliography 2124
Forcing over Models of Determinacy 2125
Iterations 2127
Pmax 2134
Sequences of Models and Countable Closure 2138
Generalized Iterability 2141
The Basic Analysis 2149
psiAC and the Axiom of Choice 2155
Maximality and Minimality 2156
Larger Models 2163
Omega-Logic 2166
Variations 2171
Variations for NSomega1 2171
Conditional Variations for Sigma2 sentences 2175
Bibliography 2178

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 two