Handbook of the History of Logic. Volume 06: Sets and Extensions in the Twentieth Century

Cover......Page 1
Title Page......Page 5
Contents......Page 7
Preface......Page 9
Contributors......Page 13
Set Theory from Cantor to Cohen......Page 15
1.1 Real numbers and countability......Page 16
1.2 Continuum Hypothesis and transfinite numbers......Page 19
1.3 Diagonalization and cardinal numbers......Page 22
2.1 Axiom of Choice and axiomatization......Page 26
2.2 Logic and paradox......Page 31
2.3 Measure, category, and Borel hierarchy......Page 34
2.4 Hausdorff and functions......Page 36
2.5 Analytic and projective sets......Page 40
2.6 Equivalences and consequences......Page 42
3.1 Ordinals and Replacement......Page 46
3.2 Well-foundedness and the cumulative hierarchy......Page 47
3.3 First-order logic and extensionalization......Page 50
3.4 Relative consistency......Page 52
3.5 Combinatorics......Page 58
3.6 Model-theoretic methods......Page 61
4.1 Forcing......Page 65
4.2 Envoi......Page 70
Acknowledgements......Page 71
Bibliography......Page 72
1 Introduction......Page 87
2 Hilbert's address......Page 88
3 Lusin's Second Continuum Hypothesis......Page 93
4 The Continuum Hypothesis......Page 97
5 Cardinal invariants of the continuum associated with convergence rates......Page 103
6 Cardinal invariants of measure and category......Page 121
7 What forcing arguments reveal about the continuum......Page 127
8 The Baire Category Theorem and Martin's Axiom......Page 135
9 Cardinal invariants of the continuum associated with βℕ\ℕ......Page 142
10 Epilogue......Page 148
Bibliography......Page 150
Infinite Combinatorics......Page 159
1 Introduction......Page 160
1.1 Overview of the history of orderings......Page 162
1.2 Overview of the history of Ramsey theory......Page 167
2 1900-1930: Beginnings......Page 170
2.1 Hausdorff......Page 173
2.2 1910–1920: Emerging schools......Page 177
2.3 1920–1930: Early structural results......Page 182
2.4 Ramsey and van der Waerden......Page 189
3 1930-1940: Early ramifications......Page 191
3.1 Extensions......Page 193
3.2 Erdős and Rado......Page 195
3.3 Jones......Page 198
3.4 Kurepa......Page 200
4 1940-1950: Pioneering partition results......Page 207
4.1 Combinatorial tools......Page 208
4.2 Ordered sets and their structure......Page 209
4.3 Around Suslin’s Problem......Page 214
4.4 Ramsey theory......Page 218
5 1950-1960: Foundation of the partition calculus......Page 224
5.1 The partition calculus......Page 226
5.2 Applications of Suslin lines......Page 235
5.3 Ordered sets, structure and mappings......Page 238
5.4 The Regressive Function Theorem......Page 239
6 1960-1970: Forcing, trees and partitions......Page 241
6.1 The Halpern-Läuchli Theorem......Page 245
6.2 Countable height trees......Page 247
6.3 Suslin’s Problem revisited......Page 251
6.4 Martin’s Axiom and diamond principles......Page 255
6.5 More on uncountable trees......Page 259
6.6 Transversals and decidability......Page 262
6.7 Partition calculus classics......Page 263
6.8 Infinitary partition relations......Page 267
7 1970-1980: Structures and forcing......Page 272
7.1 Combinatorial principles......Page 273
7.2 Transversals and cardinal arithmetic......Page 276
7.3 Partition relations on cardinals and ordinals......Page 277
7.4 Ramsey theory for trees......Page 286
7.5 Hindman’s Finite Sums Theorem......Page 287
7.6 Infinitary partition relations......Page 290
7.7 Structure of trees......Page 291
7.8 Linear and quasi-orders......Page 297
8 1980-1990: Codifications and extensions......Page 302
8.1 Set-theoretic topology......Page 304
8.2 Partition relations......Page 306
8.3 Structural partition relations......Page 311
8.5 Tree results......Page 316
8.7 Other combinatorial results......Page 322
9 1990-2000: A sampling......Page 324
9.1 Partition calculus results......Page 325
9.2 Linear and partial orders......Page 334
9.3 Trees......Page 335
9.4 Combinatorial principles......Page 337
10 Postscript......Page 338
Bibliography......Page 342
1.1 Beginnings......Page 373
1.2 Model-theoretic methods......Page 377
2.1 Cohen......Page 381
2.2 Solovay and forcing......Page 383
2.3 0^# and L[U]......Page 388
2.4 Jensen and constructibility......Page 389
3.1 Large large cardinals......Page 392
3.2 Determinacy......Page 394
3.3 Elaborations......Page 397
3.4 Silver’s theorem and covering......Page 401
3.5 Forcing consistency results through the 1970s......Page 405
4.1 Into the 1980s......Page 408
4.2 Reflecting stationary sets......Page 412
4.3 Consistency of determinacy......Page 415
4.4 Into the 1990s......Page 417
4.5 Ideals......Page 421
Bibliography......Page 423
1 Introduction......Page 429
1.1 Constructibility......Page 430
1.2 Large cardinals......Page 431
2.2 What a measurable cardinal says about L......Page 433
2.3 L[U]......Page 435
2.4 More measurable cardinals......Page 436
3 Fine structure and the covering lemma in L......Page 439
3.1 The Singular Cardinal Hypothesis and the covering lemma......Page 443
4.1 Up to one measurable cardinal......Page 445
4.2 More measures......Page 449
5.1 Moving beyond measurable cardinals......Page 453
5.2 Fine structural extender models......Page 461
5.3 L[E] as a core model......Page 462
Bibliography......Page 466
1 Introduction......Page 471
2 Early developments......Page 474
2.1 Regularity properties......Page 476
2.2 Definability......Page 477
2.3 The Axiom of Determinacy......Page 480
3 Reduction and scales......Page 482
3.1 Reduction, separation, norms and prewellorderings......Page 483
3.2 Scales......Page 485
3.3 The game quantifier......Page 487
3.4 Partially playful universes......Page 488
3.5 Wadge degrees......Page 489
4 Partition properties and the projective ordinals......Page 490
4.1 Θ, the Coding Lemma and the projective ordinals......Page 491
4.2 Partition properties and ultrafilters......Page 493
4.3 Cardinals, uniform indiscernibles and the projective ordinals......Page 494
5.1 Measurable cardinals......Page 496
5.2 Borel determinacy......Page 498
5.4 Larger cardinals......Page 499
6.1 AD⁺ and AD_ℝ......Page 505
6.2 Long games......Page 507
6.3 Forcing over models of determinacy......Page 509
6.4 Determinacy from its consequences......Page 510
6.5 Determinacy from other statements......Page 511
Bibliography......Page 513
Singular Cardinals: From Hausdorff's Gaps to Shelah's pcf Theory......Page 523
1 Introduction......Page 524
2 The beginning: Hausdorff's work......Page 533
3 Early occurrences of singular cardinals in mathematics......Page 536
3.1 Singular cardinals in topology: the work of Alexandroff and Urysohn......Page 537
3.2 More topology in products: Rudin’s space......Page 540
3.3 The Czech school’s investigations of the algebra P_{<μ}(μ)......Page 542
3.4 The Erdős-Rado work in the partition calculus and the Erdős-Hechler work on MAD families over a singular......Page 543
3.5 Singular cardinal compactness and Whitehead’s problem......Page 545
4 The arithmetic of singular cardinals......Page 546
4.1 Annus Mirabilis: the development of singular cardinal arithmetic in 1974......Page 551
5 Shelah's pcf theory......Page 553
5.1 The revised GCH above ℶ_ω......Page 561
6.2 Scott’s theorem......Page 564
6.3 Generic sequences: Prikry’s discovery......Page 565
8 Summary and concluding remarks......Page 567
Bibliography......Page 568
1 Introduction......Page 573
2.1 Simple type theory......Page 576
2.2 The original system of Zermelo......Page 578
2.3 The relationship between these systems. Mac Lane set theory......Page 579
2.4 Mac Lane or Zermelo set theory as an alternative set theory......Page 580
3.1 General considerations......Page 582
3.2 Von Neumann-Gödel-Bernays and Kelley-Morse set theory......Page 583
3.3 Ackermann set theory......Page 587
3.4 A pocket set theory......Page 592
4.1 Weak extensionality and ZFA......Page 596
4.2 Aczel’s AFA......Page 597
4.3 Boffa’s axiom......Page 599
5.1 Stratified comprehension......Page 600
5.2 New Foundations with urelements......Page 602
5.3 Peculiarities of NF......Page 612
5.4 Extensional fragments of New Foundations......Page 615
5.5 Reflections on New Foundations and ZF......Page 616
6.1 Positive set theory from the Fregean notion of set......Page 625
6.2 Positive set theory seen from the Cantorian point of view......Page 628
6.3 Topological set theory......Page 631
6.4 A development of mathematics in GPK^+_∞......Page 633
7.1 Nonstandard analysis......Page 635
7.2 Nelson’s IST......Page 636
7.3 Vopĕnka’s alternative set theory......Page 637
8.1 Double extension set theory......Page 640
8.2 Zermelo set theory with an elementary embedding......Page 641
9 Conclusions......Page 642
Bibliography......Page 643
1 The origins of type theory......Page 647
2 Critiquing ramified types......Page 654
2 Church's version of the simple theory of types......Page 657
4 Types vs. sets......Page 659
5.1 Cartesian closed categories and the typed λ-calculus......Page 661
5.2 Logical languages and local set theories......Page 665
5.3 Logic in a local set theory......Page 668
5.4 Set theory in a local language......Page 669
5.5 Interpreting a local language in a topos: the soundness and completeness theorems......Page 671
5.6 Every topos is linguistic: the equivalence theorem......Page 674
5.7 Translations of local set theories......Page 675
5.8 Classicality and the choice principle......Page 677
5.9 Characterisation of Set......Page 681
6 New forms of type theory and the doctrine of “propositions as types”......Page 683
Appendix: Basic concepts of category theory......Page 691
Bibliography......Page 699
The History of Categorical Logic: 1963-1977......Page 703
1.1 Category theory: its origins......Page 705
1.2 Category theory from 1945 until 1963......Page 709
2 Launching the program: 1963-1969......Page 711
2.1 Basic principles......Page 712
2.2 Lawvere’s thesis: 1963......Page 715
2.3 The elementary theory of the category of sets......Page 719
2.4 Categorical logic: the program......Page 721
3.1 Elementary topos theory: 1969–1970......Page 730
4 Focusing on first-order logic......Page 739
4.1 Volger’s work......Page 743
4.2 The Montreal school......Page 745
4.3 The background......Page 748
4.4 From logical theories to categories......Page 754
4.5 Algebraic logic: from regular to Boolean categories......Page 765
4.6 Constructing theories from categories......Page 769
4.7 Building bridges......Page 774
4.8 Classifying topos and generic model of a theory......Page 780
4.9 Geometric logic......Page 784
5 Higher-order logic and toposes......Page 785
5.1 Interpreting higher-order logic in toposes......Page 786
5.2 Fourman’s approach......Page 791
5.3 Revising logic: the debate......Page 793
6 The method of forcing in toposes: Kripke-Joyal semantics......Page 797
7 Fibred categories and logic......Page 799
8 The Durham meeting......Page 805
Bibliography......Page 807
1 Introduction......Page 815
2a Propositional functions......Page 819
2b Ramified types......Page 821
2c Substitution in RTT......Page 823
2d Logical truth for RTT in Tarski’s style......Page 825
3 Kripke's Theory of Truth KTT......Page 826
4a RTT embedded in KTT......Page 828
4b The restrictiveness of Russell’s theory......Page 834
4c Orders and types......Page 835
5 The Nuprl and Martin-Löf type theories......Page 836
5a A fragment of Nuprl in PTS style......Page 837
5b Orders in Nuprl......Page 840
6 Computational Type Theory CTT......Page 841
6a Origins......Page 842
6b Philosophical issues......Page 843
6c Computation and data types......Page 844
6d Elements of computational type theory......Page 846
7 Conclusions......Page 855
Bibliography......Page 856
Index......Page 861
A......Page 862
B......Page 863
C......Page 864
D......Page 865
F......Page 866
H......Page 867
J......Page 868
K......Page 869
M......Page 870
N......Page 871
P......Page 872
Q......Page 873
R......Page 874
S......Page 875
T......Page 877
V......Page 878
Z......Page 879