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The evolution of large cardinal axioms in set theory

  • A. Kanamori
  • M. Magidor
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 669)

Keywords

Regular Cardinal Large Cardinal Measurable Cardinal Elementary Embedding Compact Cardinal 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Bibliography

  1. S. Aanderaa [1973] Inductive definitions and their closure ordinals, in: J.E. Fenstad and P. Hinman,eds., Generalized Recursion Theory (North Holland, Amsterdam) 207–220.Google Scholar
  2. F.G. Abramson, L.A. Harrington, E.M. Kleinberg, and W.S. Zwicker [1977] Flipping properties: a unifying thread in the theory of large cardinals, Annals Math. Logic 12(1), 37–58.MathSciNetzbMATHCrossRefGoogle Scholar
  3. P. Aczel and W. Richter [1972] Inductive definitions and analogues of large cardinals, in: Conference in Mathematical Logic, London, 1970, Lecture Notes in Math. 255 (Springer, Berlin) 1–9.CrossRefGoogle Scholar
  4. [1974] Inductive definitions and reflecting properties of admissible ordinals, in: J.E. Fenstad and P. Hinman,eds., Generalized Recursion Theory (North Holland, Amsterdam) 301–381.Google Scholar
  5. J.Barwise, M.Kaufmann, and M.Makkai [1978] Stationary logic, Annals Math. Logic 12(3).Google Scholar
  6. J. Baumgartner [1975] Ineffability properties of cardinals I, in: Infinite and Finite Sets, Colloquia Mathematica Societatis Janos Bolyai, 10 (dedicated to P.Erdös) (North Holland, Amsterdam) vol.1.Google Scholar
  7. [1976] A new class of order types, Annals Math. Logic 9(3), 187–222.MathSciNetzbMATHCrossRefGoogle Scholar
  8. J. Baumgartner and F.Galvin (1977) Generalized Erdös cardinals and 0#, to appear.Google Scholar
  9. J. Baumgartner, A. Hajnal, and A. Maté [1975] Weak saturation properties of ideals, in: Infinite and Finite Sets, Colloquia Mathematica Societatis Janos Bolyai, 10 (dedicated to P.Erdös) (North Holland, Amsterdam).Google Scholar
  10. J. Baumgartner, L. Harrington, and E.M. Kleinberg [1976] Adding a closed unbounded set, J. Sym. Logic 41(2), 481–2.MathSciNetzbMATHCrossRefGoogle Scholar
  11. J. Baumgartner, A. Taylor, and S. Wagon [1977] On splitting stationary subsets of large cardinals, J. Sym. Logic 42, 203–214.MathSciNetzbMATHCrossRefGoogle Scholar
  12. (1978) Ideals and partitions, to appear.Google Scholar
  13. J. Bell [1977] Boolean-valued Models and Independence Proofs in Set Theory, Oxford Logic Guides (Clarendon Press, Oxford).zbMATHGoogle Scholar
  14. M. Benda and J. Ketonen [1974] On regularity of ultrafilters, Israel J. of Math. 17, 231–240.MathSciNetzbMATHCrossRefGoogle Scholar
  15. W. Boos [1975] Lectures on large cardinal axioms, in: Logic Conference, Kiel 1974, Lecture Notes in Math. 499 (Springer, Berlin) 25–88.Google Scholar
  16. L. Bukovsky [1965] The continuum problem and powers of alephs, Commentationes Math. Univ. Caolinae 6, 181–197.MathSciNetzbMATHGoogle Scholar
  17. [1976] Changing cofinality to Open image in new window, in: Set Theory and Hierarchy Theory, Lecture Notes in Math. 537 (Springer, Berlin) 37–49.Google Scholar
  18. E.Bull (1976) Successive large cardinals, to appear, and M.I.T. Ph.D. Thesis.Google Scholar
  19. C.C. Chang [1971] Sets constructible using Lκκ, in: D.S. Scott,ed., Axiomatic Set Theory, Proc. Symp. Pure Math. 13(1) (Amer. Math. Soc., Providence, R.I.) 1–8.CrossRefGoogle Scholar
  20. C.C. Chang and H.J. Keisler [1973] Model Theory (North Holland, Amsterdam).zbMATHGoogle Scholar
  21. G.V. Chudnovsky and D.V. Chudnovsky [1971] Regularnye i ubyvajusche nepolnye ultrafiltry, Dokl. Akad. Nauk U.S.S.R. 198, 779–782.Google Scholar
  22. W.W. Comfort and S. Negrepontis [1974] The Theory of Ultrafilters (Springer, Berlin).zbMATHCrossRefGoogle Scholar
  23. Morton Davis [1964] Infinite games with perfect information, in: Advances in Game Theory, Annals of Math. Studies 52 (Princeton, N.J.) 85–101.Google Scholar
  24. K.J. Devlin [1973] Some weak versions of large cardinal axioms, Annals Math. Logic 5, 291–325.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [1973a] Aspects of Constructibility, Lecture Notes in Math. 354 (Springer, Berlin).zbMATHGoogle Scholar
  26. [1974] Some remarks on changing cofinalities, J. Sym. Logic 39, 27–30.MathSciNetzbMATHCrossRefGoogle Scholar
  27. [1975] Indescribability properties and small large cardinals, in: Logic Conference, Kiel 1974, Lecture Notes in Math. 499 (Springer, Berlin) 89–114.Google Scholar
  28. K.J. Devlin and R.B. Jensen [1975] Marginalia to a theorem of Silver, in: Logic Conference, Kiel 1974, Lecture Notes in Math. 499 (Springer, Berlin) 115–142.Google Scholar
  29. K.J. Devlin and H. Johnsbraten [1974] The Souslin Problem, Lecture Notes in Math. 405 (Springer, Berlin).zbMATHGoogle Scholar
  30. M.A. Dickmann [1976] Large Infinitary Languages (North Holland, Amsterdam).zbMATHGoogle Scholar
  31. C.DiPrisco (1976) M.I.T. Ph.D. Thesis.Google Scholar
  32. C.DiPrisco and J.Henle (1977) On the compactness of Open image in new window, to appear.Google Scholar
  33. C.DiPrisco and W.Zwicker (1977) Flipping properties and supercompact cardinals, to appear.Google Scholar
  34. T.Dodd and R.B.Jensen (1976) The core model, circulated notes.Google Scholar
  35. F.R. Drake [1974] Set Theory (North Holland, Amsterdam).zbMATHGoogle Scholar
  36. W.B. Easton [1964] Powers of regular cardinals, Annals Math. Logic 1, 139–178.MathSciNetzbMATHCrossRefGoogle Scholar
  37. A. Ehrenfeucht and A. Mostowski [1956] Models of axiomatic theories admitting automorphisms, Fund. Math. 43, 50–68.MathSciNetzbMATHGoogle Scholar
  38. P. Erdös and A. Hajnal [1958] On the structure of set mappings, Acta Math. Acad. Sci. Hung. 9, 111–131.MathSciNetzbMATHCrossRefGoogle Scholar
  39. [1962] Some remarks concerning our paper "On the structure of set mappings", Acta Math. Acad. Sci. Hung. 13, 223–226.zbMATHCrossRefGoogle Scholar
  40. [1966] On a problem of B.Jónsson, Bull. Acad. Polon. Sci. Sér. Sci. Math. Ast. Phys. 14, 61–99.zbMATHGoogle Scholar
  41. [1971] Unsolved problems in set theory, in: D.S. Scott,ed., Axiomatic Set Theory, Proc. Symp. Pure Math. 13(1) (Amer. Math. Soc., Providence R.I.) 17–48.CrossRefGoogle Scholar
  42. P. Erdös, A. Hajnal, and R. Rado [1965] Partition relations for cardinal numbers, Acta Math. Acad. Sci. Hung. 16, 93–196.MathSciNetzbMATHCrossRefGoogle Scholar
  43. P. Erdös and R. Rado [1952] Combinatorial theorems on classifications of subsets of a given set, Proc. London Math. Soc. (3)2, 417–439.MathSciNetzbMATHCrossRefGoogle Scholar
  44. [1956] A partition calculus in set theory, Bull. Amer. Math. Soc. 62, 427–489.MathSciNetzbMATHCrossRefGoogle Scholar
  45. P. Erdös and A. Tarski [1943] On families of mutually exclusive sets, Ann. Math. 44, 315–429.MathSciNetzbMATHCrossRefGoogle Scholar
  46. [1961] On some problems involving inaccessible cardinals, in: Essays on the Foundations of Mathematics (Magnes Press, Jerusalem) 50–82.Google Scholar
  47. G. Fodor [1956] Eine Bemerkung zur Theorie der regressiven Funktionen, Acta Sci. Math. (Széged) 17, 139–142.MathSciNetzbMATHGoogle Scholar
  48. [1966] On a process concerning inaccessible cardinals I, Acta Sci. Math. (Széged) 27, 111–124.MathSciNetzbMATHGoogle Scholar
  49. H. Friedman [1971] Higher set theory and mathematical practice, Annals Math. Logic 2, 326–357.MathSciNetzbMATHCrossRefGoogle Scholar
  50. L. Fuchs [1970] Infinite Abelian Groups (Academic Press, London-New York).zbMATHGoogle Scholar
  51. H. Gaifman [1967] Uniform extension operators for models and their applications, in: J. Crossley, ed., Sets, Models and Recursion Theory (North Holland, Amsterdam) 122–155.CrossRefGoogle Scholar
  52. [1967a] A generalization of Mahlo's method for obtaining large cardinal numbers, Israel J. Math. 5, 188–200.MathSciNetzbMATHCrossRefGoogle Scholar
  53. [1974] Elementary embeddings of set theory, in: T.J. Jech, ed., Axiomatic Set Theory, Proc. Symp. Pure Math. 13(2) (Amer. Math. Soc., Providence,R.I.) 33–101.CrossRefGoogle Scholar
  54. D.Gale and F.M.Stewart [1953] Infinite games with perfect information, in: Kuhn and Tucker, eds., Contributions to the Theory of Games, vol. 3, Annals of Math. Studies 28 (Princeton, N.J.) 245–266.Google Scholar
  55. F. Galvin and A. Hajnal [1975] Inequalities for cardinal powers, Ann. Math. 101, 491–498.MathSciNetzbMATHCrossRefGoogle Scholar
  56. F.Galvin, T.J.Jech, and M Magidor (1976) An ideal game, to appear in J. Sym. Logic.Google Scholar
  57. F.Galvin and K.Prikry (1976) Infinitary Jónsson algebras, to appear.Google Scholar
  58. K. Glöede [1973] Filters closed under Mahlo's and Gaifman's operation, in: Cambridge Summer School in Mathematical Logic, Lecture Notes in Math. 337 (Springer, Berlin) 495–527.CrossRefGoogle Scholar
  59. K. Gödel [1938] The consistency of the axiom of choice and of the generalized continuum hypothesis, Proc. Natl. Acad. Sci. U.S.A. 24, 556–557.zbMATHCrossRefGoogle Scholar
  60. [1946] Remarks before the Princeton Bicentennial Conference on problems in mathematics, in: M. Davis, ed., The Undecidable (Raven Press, New York) 84–88.Google Scholar
  61. [1947] What is Cantor's continuum problem?, Amer. Math. Monthly 54, 515–525.MathSciNetzbMATHCrossRefGoogle Scholar
  62. J.Green (1977) Berkeley Ph.D.Thesis.Google Scholar
  63. J. Gregory [1976] Higher Souslin trees and the generalized continuum hypothesis, J. Sym. Logic 41(3), 663–671.MathSciNetzbMATHCrossRefGoogle Scholar
  64. D.Guaspari (1973) The largest constructible II11 set, to appear.Google Scholar
  65. W.P. Hanf [1964] Incompactness in languages with infinitely long expressions, Fund. Math. 53, 309–324.MathSciNetzbMATHGoogle Scholar
  66. On a problem of Erdös and Tarski, Fund. Math. 53, 325–334.MathSciNetzbMATHGoogle Scholar
  67. W.P. Hanf and D.S. Scott [1961] Classifying inaccessible cardinals, Notices Amer. Math. Soc. 8, 445.Google Scholar
  68. L.Harrington [1973] M.I.T. Ph.D.Thesis.Google Scholar
  69. [1974] The superjump and the first recursively Mahlo ordinal, in: J.E. Fenstad and P. Hinman, eds., Generalized Recursion Theory (North Holland, Amsterdam) 43–52.Google Scholar
  70. (1975) Analytic determinacy and 0#, to appear.Google Scholar
  71. F. Hausdorff [1908] Grundzüge einer Theorie der geordneten Mengen, Math. Annalen 65, 443.MathSciNetzbMATHCrossRefGoogle Scholar
  72. E. Hewitt [1948] Rings of continuous functions I, Trans. Amer. Math. Soc. 64, 45–99.MathSciNetzbMATHCrossRefGoogle Scholar
  73. T.J. Jech [1968] θ1 can be measurable, Israel J. Math. 6, 363–367.MathSciNetzbMATHCrossRefGoogle Scholar
  74. [1971] Trees, J. Sym. Logic 36(1), 1–14.MathSciNetzbMATHCrossRefGoogle Scholar
  75. [1973] Some combinatorial problems concerning uncountable cardinals, Annals Math. Logic 5, 165–198.MathSciNetzbMATHCrossRefGoogle Scholar
  76. (1977) A game theoretic property of Boolean algebras, to appear in the proceedings of Wroclaw 1977 conference.Google Scholar
  77. T.J.Jech, M.Magidor, W.Mitchell, and K.Prikry (1977) On precipitous ideals, to appear.Google Scholar
  78. T.J.Jech and K Prikry (1977) Ideals over uncountable sets: applications of almost disjoint functions and generic ultrapowers, to appear.Google Scholar
  79. R.B. Jensen [1972] The fine structure of the constructible hierarchy, Annals Math. Logic 4, 229–308.MathSciNetzbMATHCrossRefGoogle Scholar
  80. (1975) Coding the universe in a real, circulated notes.Google Scholar
  81. R.B.Jensen and B.Koppelberg (1977) Regularity of ultrafilters, to appear.Google Scholar
  82. R.B.Jensen and K.Kunen (1971) Some combinatorial properties of L and V, circulated notes.Google Scholar
  83. R.B. Jensen and R.M. Solovay [1970] Some applications of almost disjoint sets, in: Y. Bar-Hillel, ed., Mathe-Logic and Foundations of Set Theory (North Holland, Amsterdam) 84–104.Google Scholar
  84. A. Kanamori [1976] Weakly normal filters and irregular ultrafilters, Trans. Amer. Math. Soc. 220, 393–399.MathSciNetzbMATHCrossRefGoogle Scholar
  85. [1977] Ultrafilters over a measurable cardinal, Annals Math. Logic 11, 315–356.MathSciNetzbMATHGoogle Scholar
  86. [1978] On Vopěnka's and related principles, to appear in the proceedings of Wroclaw 1977 conference.Google Scholar
  87. C. Karp [1964] Languages with Expressions of Infinite Length (North Holland, Amsterdam).zbMATHGoogle Scholar
  88. A. Kechris [1973] Measure and category in effective descriptive set theory, Annals Math. Logic 5, 337–384.MathSciNetzbMATHCrossRefGoogle Scholar
  89. [1974] On projective ordinals, J. Sym. Logic 39(2), 269–282.MathSciNetzbMATHCrossRefGoogle Scholar
  90. [1975] The theory of countable analytical sets, Trans Amer. Math. Soc. 202, 259–297.MathSciNetzbMATHCrossRefGoogle Scholar
  91. (1977) AD and infinite exponent partition relations, manuscript.Google Scholar
  92. H.J. Keisler [1971] Model Theory for Infinitary Logic (North Holland, Amsterdam).zbMATHGoogle Scholar
  93. H.J. Keisler and A. Tarski [1964] From accessible to inaccessible cardinals, Fund. Math. 53, 225–308.MathSciNetzbMATHGoogle Scholar
  94. J. Ketonen [1972] Strong compactness and other cardinal sins, Annals Math. Logic 5, 47–76.MathSciNetzbMATHCrossRefGoogle Scholar
  95. [1976] Non-regular ultrafilters and large cardinals, Trans. Amer. Math. Soc. 224, 61–73.MathSciNetzbMATHCrossRefGoogle Scholar
  96. E.M. Kleinberg [1970] Strong partition properties for infinite cardinals, J. Sym. Logic 35, 410–428.MathSciNetzbMATHCrossRefGoogle Scholar
  97. [1973] Infinitary combinatorics, in: Cambridge Summer School in Mathematical Logic, Lecture Notes in Math. 337 (Springer, Berlin) 361–418.CrossRefGoogle Scholar
  98. [1973a] Rowbottom cardinals and Jónsson cardinals are almost the same, J. Sym. Logic 38(3) 423–427.MathSciNetzbMATHCrossRefGoogle Scholar
  99. (1973) The equiconsistency of two large cardinal axioms, to appear in Fund. Math.Google Scholar
  100. [1977] Infinitary Combinatorics and the Axiom of Determinateness, Lecture Notes in Math. 612 (Springer, Berlin).zbMATHGoogle Scholar
  101. D.W. Kueker [1977] Countable approximations and Löwenheim-Skolem theorems, Annals Math. Logic 11(1), 57–103.MathSciNetzbMATHCrossRefGoogle Scholar
  102. K.Kunen [1968] Inaccessibility properties of cardinals, Stanford Ph.D.Thesis.Google Scholar
  103. [1970] Some applications of iterated ultrapowers in set theory, Annals Math. Logic 1, 179–227.MathSciNetzbMATHCrossRefGoogle Scholar
  104. [1970a] On the compactification of the integers, Notices Amer. Math. Soc. 17,299.Google Scholar
  105. [1971] Elementary embeddings and infinitary combinatorics, J. Sym. Logic 36, 407–413.MathSciNetzbMATHCrossRefGoogle Scholar
  106. [1971a] On the GCH at measurable cardinals, in: R.O. Gandy and C.E.M. Yates, eds., Logic Colloquium '69 (North Holland, Amsterdam) 107–110.CrossRefGoogle Scholar
  107. [1973] A model for the negation of the axiom of choice, in: Cambridge Summer School in Mathematical Logic, Lecture Notes in Math. 337 (Springer, Berlin) 489–494.CrossRefGoogle Scholar
  108. (1974) Saturated ideals, to appear in J. Sym. Logic.Google Scholar
  109. [1976] Some points in βN, Proc. Cambridge Phil. Soc. 80, 385–398.MathSciNetzbMATHCrossRefGoogle Scholar
  110. K. Kunen and J.B. Paris [1971] Boolean extensions and measurable cardinals, Annals Math. Logic 2, 359–378.MathSciNetzbMATHCrossRefGoogle Scholar
  111. G. Kurepa [1935] Ensembles ordonnés et ramifiés, Publ. Math. Univ. Belgrade 4, 1–138.zbMATHGoogle Scholar
  112. [1936] L'hypothèse de ramification, Comptes Rendus, Hebdomadaires des Séances de l'Academie des Sciences, Séries A et B, vol.202, 185–187.zbMATHGoogle Scholar
  113. R.Laver (1974) Making the supercompactness of κ indestructible under κ directed closed forcing, to appear.Google Scholar
  114. (1976) A saturation property of ideals and the partition relation κ → (κ,α)2.Google Scholar
  115. A. Lévy [1960] A generalization of Gödel's notion of constructibility, J. Sym. Logic 25, 147–155.zbMATHCrossRefGoogle Scholar
  116. [1965] A hierarchy of formulas in set theory, Mem. Amer. Math. Soc. 57.Google Scholar
  117. [1971] The sizes of the indescribable cardinals, in: D.S. Scott, ed., Axiomatic Set Theory, Proc. Symp. Pure Math. 13(1) (Amer. Math. Soc., Providence, R.I.) 205–218.CrossRefGoogle Scholar
  118. A. Lévy and R.M. Solovay [1967] Measurable cardinals and the continuum hypothesis, Israel J. Math. 5, 234–248.MathSciNetzbMATHCrossRefGoogle Scholar
  119. M. Magidor [1971] On the role of supercompact and extendible cardinals in logic, Israel J. Math. 10, 147–157.MathSciNetzbMATHCrossRefGoogle Scholar
  120. [1974] Combinatorial characterization of supercompact cardinals, Proc. Amer. Math. Soc. 42(1).Google Scholar
  121. (1975) Changing cofinalities of cardinals, to appear.Google Scholar
  122. [1976] How large is the first strongly compact cardinal?, Annals Math. Logic 10, 33–57.MathSciNetzbMATHCrossRefGoogle Scholar
  123. [1977] On the singular cardinals problem I, Israel J. of Math. 28(1), 1–31.MathSciNetzbMATHCrossRefGoogle Scholar
  124. [1977a] Chang's conjecture and powers of singular cardinals, J. Sym. Logic 42(2), 272–276.MathSciNetzbMATHCrossRefGoogle Scholar
  125. (1977) On the existence of non-regular ultrafilters and cardinality of ultrapowers, to appear.Google Scholar
  126. [1978] On the singular cardinals problem II, to appear in Ann. Math.Google Scholar
  127. P. Mahlo [1911] Uber lineare transfinite Mengen, Ber. Verhandl. Sächs. Akad. Wiss. Leipzig, Math.-Naturw. Kl. 63, 187–225.zbMATHGoogle Scholar
  128. [1912] Zur Theorie und Andwendung der ρ0-Zahlen, Ber. Verhandl. Sächs. Akad. Wiss. Leipzig, Math.-Naturw. Kl. 64, 108–112.Google Scholar
  129. [1913] Zur Theorie und Andwendung der ρ0-Zahlen II, Ber. Verhandl. Sächs. Akad. Wiss. Leipzig, Math.-Naturw. Kl. 64, 268–282.Google Scholar
  130. R. Mansfield [1970] Perfect subsets of definable sets of real numbers, Pacific J. Math. 35, 451–457.MathSciNetzbMATHCrossRefGoogle Scholar
  131. D.A. Martin [1970] Measurable cardinals and analytic games, Fund. Math. 66, 287–291.MathSciNetzbMATHGoogle Scholar
  132. (1971) Projective sets and cardinal numbers, to appear.Google Scholar
  133. [1975] Borel Determinacy, Ann. Math. 102, 363–371.MathSciNetzbMATHCrossRefGoogle Scholar
  134. D.A. Martin and R.M. Solovay [1969] A basis theorem for Σ31 sets of reals, Ann. Math. 89(2), 138–160.MathSciNetzbMATHCrossRefGoogle Scholar
  135. [1970] Internal Cohen extensions, Annals Math. Logic 2, 143–178.MathSciNetzbMATHCrossRefGoogle Scholar
  136. A.R.D. Mathias [1977] Happy families, Annals Math. Logic 12(1), 59–111.MathSciNetzbMATHCrossRefGoogle Scholar
  137. T.K. Menas [1974] On strong compactness and supercompactness, Annals Math. Logic 7, 327–359.MathSciNetzbMATHCrossRefGoogle Scholar
  138. [1975] A combinatorial property of pκλ, J. Sym. Logic 41(1), 225–233.MathSciNetGoogle Scholar
  139. [1976] Consistency results concerning supercompactness, Trans. Amer. Math. Soc. 223, 61–91.MathSciNetzbMATHCrossRefGoogle Scholar
  140. A.Miller [1977] Berkeley Ph.D.Thesis.Google Scholar
  141. W. Mitchell [1972] Aronszajn trees and the independence of the transfer property, Annals Math. Logic 5, 21–46.MathSciNetzbMATHCrossRefGoogle Scholar
  142. [1974] Sets constructible from sequences of ultrafilters, J. Sym. Logic 39, 57–66.MathSciNetzbMATHCrossRefGoogle Scholar
  143. Y.N. Moschovakis [1970] Determinacy and prewellorderings of the continuum, in: Y. Bar-Hillel, ed., Mathematical Logic and Foundations of Set Theory (North Holland, Amsterdam) 24–62.Google Scholar
  144. [1973] Analytical definability in a playful universe, in: Suppes et al, eds., Proceedings of Third International Congress for Logic, Methodology and Philosophy of Science, Bucharest 1971 (North Holland, Amsterdam) 77–85.CrossRefGoogle Scholar
  145. J. Mycielski [1964] On the axiom of determinateness, Fund. Math. 53, 205–224.MathSciNetzbMATHGoogle Scholar
  146. J. Mycielski and H. Steinhaus [1962] On a mathematical axiom contradicting the axiom of choice, Bull. Acad. Polon. Sci. Sér. Sci. Math. Ast. Phys. 10, 67–71.MathSciNetzbMATHGoogle Scholar
  147. J. Mycielski and S. Swierczkowski [1964] The Lebesgue measurability and the axiom of determinateness, Fund. Math. 54, 67–71.MathSciNetzbMATHGoogle Scholar
  148. J. Mycielski, S. Swierczkowski, and A. Zieba [1956] On infinite positional games, Bull. Acad. Polon. Sci. Sér. Sci. Math. Ast. Phys. 4, 485–488.MathSciNetzbMATHGoogle Scholar
  149. J. Mycielski and A. Zieba [1955] On infinite positional games, Bull. Acad. Polon. Sci. Sér. Sci. Math. Ast. Phys. 3, 133–136.MathSciNetzbMATHGoogle Scholar
  150. J.R. Myhill and D.S. Scott [1971] Ordinal definability, in: D.S. Scott, ed., Axiomatic Set Theory, Proc. Symp. Pure Math. 13(1) (Amer. Math. Soc., Providence, R.I.) 271–278.CrossRefGoogle Scholar
  151. J.B. Paris [1972] ZF ⊢ Σ40 Determinateness, J.Sym. Logic 37, 661–667.MathSciNetzbMATHCrossRefGoogle Scholar
  152. K. Prikry [1970] Changing measurable into accessible cardinals, Dissertationes Mathematicae (Rozprawy Matematyczne) 68, 5–52.MathSciNetzbMATHGoogle Scholar
  153. [1971] On a problem of Gillman and Keisler, Annals Math. Logic 2, 179–187.MathSciNetzbMATHCrossRefGoogle Scholar
  154. [1973] On descendingly complete ultrafilters, in: Cambridge Summer School in Mathematical Logic, Lecture Notes in Math. 337 (Springer, Berlin) 459–488.CrossRefGoogle Scholar
  155. [1974] Ideals and powers of cardinals, to appear.Google Scholar
  156. W.N. Reinhardt [1974] Remarks on reflection principles, large cardinals, and elementary embeddings, in: T.J. Jech, ed., Axiomatic Set Theory, Proc. Symp. Pure Math. 13(2) (Amer. Math. Soc., Providence, R.I.) 189–205.CrossRefGoogle Scholar
  157. G.E. Reyes [1972] \(L_{\omega _1 \omega } \) is enough: a reduction theorem for some infinitary languages, J. Sym. Logic 37, 705–710.MathSciNetCrossRefGoogle Scholar
  158. W. Richter [1971] Recursively Mahlo ordinals and inductive definitions, in: R.O. Gandy and C.E.M. Yates, eds., Logic Colloquium '69 (North Holland, Amsterdam), 273–288.CrossRefGoogle Scholar
  159. F. Rowbottom [1964] Some strong axioms of infinity incompatible with the axiom of constructibility, Madison Ph.D. Thesis; published as: Annals Math. Logic 3(1971) 1–44.MathSciNetzbMATHGoogle Scholar
  160. G.E. Sacks [1976] Countable admissible ordinals and hyperdegrees, Advances in Math. 20(2), 213–262.MathSciNetzbMATHCrossRefGoogle Scholar
  161. D.S. Scott [1961] Measurable cardinals and constructible sets, Bull. Acad. Polon. Sci. Sér. Math. Ast. Phys. 7, 145–149.zbMATHGoogle Scholar
  162. S. Shelah [1975] A compactness theorem for singular cardinals, free algebras, Whitehead problem and transversals, Israel J. Math. 21(4), 319–349.MathSciNetzbMATHCrossRefGoogle Scholar
  163. (1977) Jónsson groups, to appear in proceedings of Oxford 1977 conference (North Holland, Amsterdam).Google Scholar
  164. J.R. Shoenfield [1971] Unramified forcing, in: D.S. Scott, ed., Axiomatic Set Theory, Proc. Symp. Pure Math. 13(1) (Amer. Math. Soc., Providence, R.I.) 357–381.CrossRefGoogle Scholar
  165. W. Sierpinski and A. Tarski [1930] Sur une propriété caractéristique des nombres inaccessibles, Fund. Math. 15, 292–300.zbMATHGoogle Scholar
  166. J.H. Silver [1966] Some applications of model theory in set theory, Berkeley Ph.D.Thesis, published as: Annals Math. Logic 3(1970), 45–110.MathSciNetzbMATHGoogle Scholar
  167. [1970] A large cardinal in the constructible universe, Fund. Math. 69, 93–100.MathSciNetzbMATHGoogle Scholar
  168. [1971] The consistency of the GCH with the existence of a measurable cardinal, in: D.S. Scott, ed., Axiomatic Set Theory, Proc. Symp. Pure Math. 13(1) (Amer. Math. Soc., Providence, R.I.) 391–395.CrossRefGoogle Scholar
  169. [1971a] The independence of Kurepa's conjecture and two-cardinal conjectures in model theory, in: D.S. Scott, ed., Axiomatic Set Theory, Proc. Symp. Pure Math. 13(1) (Amer. Math. Soc., Providence, R.I.) 383–390.CrossRefGoogle Scholar
  170. [1971b] Measurable cardinals and Δ31 well-orderings, Ann. Math. 94, 414–446.MathSciNetzbMATHCrossRefGoogle Scholar
  171. [1974] Indecomposable ultrafilters and 0#, in: L. Henkin et al, eds., Proceedings of the Tarski Symposium, Proc. Symp. Pure Math. 25 (Amer. Math. Soc., Providence, R.I.).Google Scholar
  172. [1974a] On the singular cardinals problem, in: Proceedings of the International Congress of Matheamticians, Vancouver 1974, vol. 1, 265–268.MathSciNetGoogle Scholar
  173. E.C. Smith and A. Tarski [1957] Higher degrees of distributivity and completeness in Boolean algebras, Trans. Amer. Math. Soc. 84, 230–257.MathSciNetzbMATHCrossRefGoogle Scholar
  174. R.M. Solovay [1967] A non-constructible Δ31 set of integers, Trans. Amer. Math. Soc. 127, 58–75.MathSciNetGoogle Scholar
  175. [1969] The cardinality of Σ21 sets of reals, in: J.J. Bullof, T.C. Holyoke, and S. Wilthan, eds., Foundations of Mathematics, Symposium papers commemorating the sixtieth birthday of Kurt Gödel (Springer, Berlin) 59–73.Google Scholar
  176. [1970] A model of set theory in which every set of reals is Lebesgue measurable, Ann. Math. 92, 1–56.MathSciNetzbMATHCrossRefGoogle Scholar
  177. [1971] Real-valued measurable cardinals, in: D.S. Scott, ed., Axiomatic Set Theory, Proc. Symp. Pure Math. 13(1) (Amer. Math. Soc., Providence, R.I.) 397–428.CrossRefGoogle Scholar
  178. [1974] Strongly compact cardinals and the G.C.H., in: L. Henkin et al, eds., Proceedings of the Tarski Symposium, Proc. Symp. Pure Math. 25 (Amer. Math. Soc., Providence, R.I.) 365–372.CrossRefGoogle Scholar
  179. R.M. Solovay, W.N. Reinhardt, and A. Kanamori [1978] Strong axioms of infinity and elementary embeddings, Annals Math. Logic 13(1), 73–116.MathSciNetzbMATHCrossRefGoogle Scholar
  180. R.M. Solovay and S. Tennenbaum [1971] Iterated Cohen extensions and Souslin's problem, Ann. Math. 94, 201–245.MathSciNetzbMATHCrossRefGoogle Scholar
  181. E. Specker [1951] Sur un problème de Sikorski, Colloq. Math. 2, 9–12.MathSciNetzbMATHGoogle Scholar
  182. [1957] Zur Axiomatik der Mengenlehre (Fundierungs und Answahl Axiome), Zeitschrift f. math. Logic und Grundlagen der Math. 3, 173–210.MathSciNetzbMATHCrossRefGoogle Scholar
  183. M. Spector [1978] M.I.T. Ph.D.Thesis.Google Scholar
  184. J. Stern (1973) The second strongly compact cardinal, notes.Google Scholar
  185. D.H. Stewart [1966] Bristol M.Sc.Thesis.Google Scholar
  186. A. Tarski [1939] Ideale in vollständigen Mengenkörpen I, Fund. Math. 32, 45–63.zbMATHGoogle Scholar
  187. [1962] Some problems and results relevant to the foundations of set theory, in: E. Nagel, P. Suppes, and A. Tarski, eds., Logic, Methodology, and Philosophy of Science (Stanford, California) 125–135.Google Scholar
  188. A. Taylor (1977) Regularity properties of ideals and ultrafilters.Google Scholar
  189. S. Ulam [1930] Zur Masstheorie in der algemeinen Mengenlehre, Fund. Math. 16, 140–150.zbMATHGoogle Scholar
  190. [1964] Combinatorial analysis in infinite sets, SIAM Rev., 343–355.Google Scholar
  191. P. Vopěnka and K. Hrbáček [1966] On strongly measurable cardinals, Bull. Acad. Polon. Sci. Sér. Sci. Math. Ast. Phys. 14, 587–591.MathSciNetzbMATHGoogle Scholar
  192. S. Wagon (1975) The structure of precipitous ideals, to appear.Google Scholar

Copyright information

© Springer-Verlag 1978

Authors and Affiliations

  • A. Kanamori
    • 1
  • M. Magidor
    • 2
  1. 1.Harvard UniversityCambridge
  2. 2.Ben Gurion UniversityBeer-ShevaIsrael

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