Handbook Of Set Theory

Description

This handbook is the definitive compendium of the methods, results, and current initiatives in modern set theory in all its research directions. Set theory has entered its prime as an advanced and autonomous field of mathematics with foundational significance, and the expanse and variety of this handbook attests to the richness and sophistication of the subject. The chapters are written by acknowledged experts, major research figures in their areas, and they each bring to bear their experience and insights in carefully wrought, self-contained expositions. There is historical depth, elegant development, probing to the frontiers, and prospects for the future. This handbook is essential reading for the aspiring researcher, a pivotal focus for the veteran set theorist, and a massive reference for all those who want to gain a larger sense of the tremendous advances that have been made in the subject, one which first appeared as a foundation of mathematics but in the last several decades has expanded into a broad and far-reaching field with its own self-fueling initiatives. Introduction......Page 15 Cantor......Page 16 Zermelo......Page 18 First Developments......Page 20 Replacement and Foundation......Page 24 Gödel......Page 27 Infinite Combinatorics......Page 31 Definability......Page 33 Model-Theoretic Techniques......Page 36 Cohen......Page 41 Method of Forcing......Page 43 0#, L[U], and L[U]......Page 47 Constructibility......Page 50 Large Large Cardinals......Page 55 Determinacy......Page 58 Silver's Theorem and Covering......Page 62 Forcing Consistency Results......Page 66 Into the 1980s......Page 70 Consistency of Determinacy......Page 74 Later Developments......Page 77 Summaries of the Handbook Chapters......Page 82 Stationary Sets......Page 107 Closed Unbounded Sets......Page 108 Splitting Stationary Sets......Page 110 Generic Ultrapowers......Page 111 Stationary Sets in Generic Extensions......Page 113 Reflecting Stationary Sets......Page 114 A Hierarchy of Stationary Sets......Page 117 Canonical Stationary Sets......Page 118 Full Reflection......Page 119 kappa+-saturation......Page 120 Precipitousness......Page 123 Closed Unbounded Sets in PkappaA......Page 124 Splitting Stationary Sets......Page 127 Saturation......Page 128 Proper Forcing......Page 129 Projective and Cohen Boolean Algebras......Page 130 Reflection Principles......Page 132 Stationary Tower Forcing......Page 135 Bibliography......Page 136 Partition Relations......Page 143 Introduction......Page 144 Basic Definitions......Page 146 Ramsey's Theorem......Page 148 Ramification Arguments......Page 149 Partition Relations and Submodels......Page 151 Overview......Page 154 More Elementary Submodels......Page 157 The Balanced Generalization......Page 158 The Unbalanced Generalization......Page 161 The Baumgartner-Hajnal Theorem......Page 166 The Milner-Rado Paradox and Omega(kappa)......Page 174 Shelah's Theorem for Infinitely Many Colors......Page 176 Singular Cardinal Resources......Page 179 Successors of Weakly Compact Cardinals......Page 181 Successors of Singular Cardinals......Page 185 Some History......Page 190 Small Counterexamples......Page 191 A Positive Countable Partition Relation......Page 201 Representation......Page 202 Node Labeled Trees......Page 206 Game......Page 208 Uniformization......Page 210 Triangles......Page 213 Free Sets......Page 218 Bibliography......Page 223 Coherent Sequences......Page 228 The Space of Countable Ordinals......Page 230 Subadditive Functions......Page 237 Steps and Coherence......Page 244 The Trace and the Square-Bracket Operation......Page 249 A Square-Bracket Operation on a Tree......Page 256 Special Trees and Mahlo Cardinals......Page 257 The Weight Function on Successor Cardinals......Page 261 The Number of Steps......Page 263 Square Sequences......Page 265 The Full Lower Trace of a Square Sequence......Page 270 Special Square Sequences......Page 273 Successors of Regular Cardinals......Page 276 Successors of Singular Cardinals......Page 280 The Oscillation Mapping......Page 286 The Square-Bracket Operation......Page 288 Unbounded Functions on Successors of Regular Cardinals......Page 294 Higher Dimensions......Page 299 Bibliography......Page 305 Borel Equivalence Relations......Page 310 Definitions......Page 311 Structure......Page 319 Anti-Structure......Page 323 Beyond Good and Evil......Page 324 Countable Borel Equivalence Relations......Page 326 The Global Structure......Page 327 Treeable Equivalence Relations......Page 334 Hyperfiniteness......Page 335 Effective Cardinality......Page 336 Classification Problems......Page 337 Universal for Polish Group Actions......Page 338 Universal for Sinfty......Page 339 E0......Page 340 Bibliography......Page 341 Proper Forcing......Page 346 Introduction......Page 347 Countable Support Iterations......Page 352 Properness and Its Iteration......Page 353 Preservation of Properness......Page 357 The 2-Chain Condition......Page 360 Equivalent Formulations......Page 363 Preservation of omegaomega-Boundedness......Page 365 Application: Non-Isomorphism of Ultrapowers......Page 369 Preservation of Unboundedness......Page 374 The Almost Bounding Property......Page 376 Application to Cardinal Invariants......Page 377 alpha-Properness......Page 382 Equivalent Definition......Page 383 Preservation of alpha-Properness......Page 384 A Coloring Problem......Page 387 Dee-Completeness......Page 392 Two-Step Iteration......Page 396 Proof of Theorem 5.17......Page 398 Simple Completeness Systems......Page 401 The Properness Isomorphism Condition......Page 403 Bibliography......Page 406 Combinatorial Cardinal Characteristics of the Continuum......Page 408 Introduction......Page 409 Growth of Functions......Page 411 Splitting and Homogeneity......Page 415 Galois-Tukey Connections and Duality......Page 421 Category and Measure......Page 430 Sparse Sets of Integers......Page 438 Forcing Axioms......Page 447 Almost Disjoint and Independent Families......Page 455 Filters and Ultrafilters......Page 459 Evasion and Prediction......Page 471 Forcing......Page 481 Countable Support Proper Iteration......Page 483 Cohen Reals......Page 485 Random Reals......Page 486 Sacks Reals......Page 488 Hechler Reals......Page 490 Mathias Reals......Page 491 Miller Reals......Page 492 Other Forcing Iterations......Page 493 Adding One Real......Page 495 Bibliography......Page 496 Invariants of Measure and Category......Page 503 Introduction......Page 504 Tukey Connections......Page 505 Inequalities Provable in ZFC......Page 507 Combinatorial Characterizations......Page 519 Cofinality of cov(J) and COV(J)......Page 530 Consistency Results and Counterexamples......Page 547 Further Reading......Page 560 Bibliography......Page 564 Constructibility and Class Forcing......Page 568 Three Problems of Solovay......Page 570 Tameness......Page 572 Examples......Page 576 Relevance......Page 580 The Coding Theorem......Page 588 The Solovay Problems......Page 599 Generic Saturation......Page 602 Further Results......Page 605 Some Open Problems......Page 613 Bibliography......Page 614 Fine Structure......Page 616 Acceptable J-structures......Page 617 The First Projectum......Page 629 Downward Extension of Embeddings......Page 632 Upward Extension of Embeddings......Page 636 Iterated Projecta......Page 642 Standard Parameters......Page 646 Solidity Witnesses......Page 649 Fine Ultrapowers......Page 652 Applications to L......Page 662 Bibliography......Page 666 Sigma Fine Structure......Page 668 Sigma Fine Structure......Page 669 Variant Fine Structures......Page 682 Sigma Ultrapowers......Page 684 Pseudo-Ultrapowers......Page 696 Global......Page 703 Defining Cnu......Page 712 Variants and Generalities on......Page 725 in Fine-Structural Inner Models......Page 731 Morasses......Page 735 Construction of Gap-1 Morasses in L......Page 737 Variants......Page 741 Bibliography......Page 744 Elementary Embeddings and Algebra......Page 748 Kunen's Bound and Axiom (I3)......Page 749 Operations on Elementary Embeddings......Page 751 Iterations of an Elementary Embedding......Page 753 Finite Quotients......Page 756 The Laver-Steel Theorem......Page 760 Counting the Critical Ordinals......Page 762 Iterated Left Division in LD-systems......Page 767 Using Elementary Embeddings......Page 770 Avoiding Elementary Embeddings......Page 771 Finite LD-Systems......Page 773 Using Elementary Embeddings......Page 776 Not Avoiding Elementary Embeddings?......Page 779 Bibliography......Page 784 Iterated Forcing and Elementary Embeddings......Page 786 Introduction......Page 787 Elementary Embeddings......Page 792 Ultrapowers and Extenders......Page 795 Large Cardinal Axioms......Page 798 Forcing......Page 800 Some Forcing Posets......Page 805 Iterated Forcing......Page 809 Building Generic Objects......Page 814 Lifting Elementary Embeddings......Page 817 Generic Embeddings......Page 820 Iteration with Easton Support......Page 822 Master Conditions......Page 825 A Technique of Magidor......Page 830 Absorption......Page 831 Transfer and Pullback......Page 839 Small Large Cardinals......Page 840 A Precipitous Ideal on omega1......Page 842 Iterated Club Shooting......Page 844 Details......Page 845 Precipitousness......Page 848 A Lower Bound......Page 851 Precipitousness for NSomega2 | Cof(omega1)......Page 852 Outline of the Proof and Main Technical Issues......Page 853 Namba Forcing, RCS Iteration and the S and I Conditions......Page 855 The Preparation Iteration......Page 857 A Warm-up for the Main Iteration......Page 858 The Main Iteration......Page 863 Precipitousness of the Non-Stationary Ideal......Page 865 Successors of Larger Cardinals......Page 866 More on Iterated Club Shooting......Page 867 More on Collapses......Page 871 Limiting Results......Page 873 Termspace Forcing......Page 876 More on Termspace Forcing and Collapsing......Page 879 Iterations with Prediction......Page 882 Altering Generic Objects......Page 888 Bibliography......Page 890 Ideals and Generic Elementary Embeddings......Page 895 Introduction......Page 897 An Overview of the Chapter......Page 898 Apology and Acknowledgments......Page 900 Some Conventions Used in the Chapter......Page 901 Basic Facts......Page 902 The Generic Ultrapower......Page 904 Disjointing Property and Closure of Ultrapowers......Page 908 Normal Ideals......Page 910 More General Facts......Page 914 Canonical Functions......Page 916 Ideals and Reflection......Page 918 Examples......Page 920 The Closed Unbounded Filter and the Nonstationary Ideal......Page 921 Natural Ideals on P(R)......Page 927 I[lambda] and Related Ideals......Page 928 Club Guessing Ideals......Page 930 Ideals of Sets Without Guessing Sequences......Page 931 Uniformization Ideals......Page 933 Induced Ideals......Page 934 General Induced Ideals......Page 937 Goodness and Self-Genericity......Page 938 A Structural Property of Saturated Ideals......Page 941 Saturation Properties......Page 942 Layered Ideals......Page 947 Projections......Page 951 Where the Ordinals Go......Page 955 A Discussion of Large Sets......Page 958 Iterating Ideals......Page 962 Generic Ultrapowers by Towers......Page 964 Using Reflection......Page 968 Chang's Conjectures, Jónsson Cardinals and Square......Page 970 Ideals and GCH......Page 972 Woodin's Theorem Showing CH Holds......Page 973 Abe's Results on SCH......Page 977 The Value of Theta......Page 978 Stationary Set Reflection......Page 981 Suslin and Kurepa Trees......Page 983 Partition Properties......Page 985 The Normal Moore Space Conjecture and Variants......Page 989 Consequences in Descriptive Set Theory......Page 990 Connections with Non-Regular Ultrafilters......Page 991 Fully Non-Regular Ultrafilters on omega2......Page 994 Graphs and Chromatic Numbers......Page 995 The Nonstationary Ideal on omega1......Page 996 Some Limitations......Page 999 Soft Limitations......Page 1000 The "Kunen Argument"......Page 1002 Saturated Ideals and Cofinalities......Page 1004 Closed Unbounded Subsets of [kappa]omega......Page 1008 Uniform Ideals on Ordinals......Page 1009 Restrictions on the Quotient Algebra......Page 1011 Yet Another Result of Kunen......Page 1017 The Matsubara-Shioya Theorem......Page 1018 The Nonstationary Ideal on [lambda]