Skip to main content
SpringerLink
Log in
SpringerLink
Log in

Ideals and Generic Elementary Embeddings

  • Chapter
  • First Online:
Handbook of Set Theory

Abstract

This chapter covers the technique of generic elementary embeddings. These embeddings are closely analogous to conventional large cardinal embeddings, the difference being that they are definable in forcing extensions of V rather than in V itself. The advantage of allowing the embeddings to be generic is that the critical points of the embeddings can be quite small, even as small as ω 1. For this reason they have many consequences for accessible cardinals, settling many classical questions of set theory.

The writing of this chapter was partially supported by grant MS-0701030 from the National Science Foundation.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 709.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 899.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 899.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. Yoshihiro Abe. Weakly normal ideals on P κ λ and the singular cardinal hypothesis. Fundamenta Mathematicae, 143(2):97–106, 1993.

    MATH  MathSciNet  Google Scholar 

  2. Uri Abraham. Aronszajn trees on \(\aleph\sb{2}\) and \(\aleph\sb{3}\) . Annals of Pure and Applied Logic, 24(3):213–230, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  3. James E. Baumgartner. Independence results in set theory. Notices of the American Mathematical Society, 25A:248–249, 1978.

    Google Scholar 

  4. James E. Baumgartner. On the size of closed unbounded sets. Annals of Pure and Applied Logic, 54(3):195–227, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  5. James E. Baumgartner and Alan D. Taylor. Saturation properties of ideals in generic extensions. I. Transactions of the American Mathematical Society, 270(2):557–574, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  6. James E. Baumgartner and Alan D. Taylor. Saturation properties of ideals in generic extensions. II. Transactions of the American Mathematical Society, 271(2):587–609, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  7. James E. Baumgartner, András Hajnal, and Stevo Todorčević. Extensions of the Erdős-Rado theorem. In Finite and Infinite Combinatorics in Sets and Logic (Banff, 1991), volume 411 of NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, pages 1–17. Kluwer Academic, Norwel, 1993.

    Google Scholar 

  8. James E. Baumgartner, Alan D. Taylor, and Stanley Wagon. On splitting stationary subsets of large cardinals. The Journal of Symbolic Logic, 42(2):203–214, 1977.

    Article  MATH  MathSciNet  Google Scholar 

  9. James E. Baumgartner, Alan D. Taylor, and Stanley Wagon. Structural properties of ideals. Dissertationes Mathematicae (Rozprawy Matematyczne), 197:95, 1982.

    MathSciNet  Google Scholar 

  10. R.H. Bing. Metrizability of topological spaces. Canadian Journal of Mathematics, 3:175–186, 1951.

    MATH  MathSciNet  Google Scholar 

  11. Douglas R. Burke. Precipitous towers of normal filters. The Journal of Symbolic Logic, 62(3):741–754, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  12. Douglas R. Burke. On a question of Abe. Fundamenta Mathematicae, 163:95–98, 2000.

    MATH  MathSciNet  Google Scholar 

  13. Douglas R. Burke and Yo Matsubara. Ideals and combinatorial principles. The Journal of Symbolic Logic, 62(1):117–122, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  14. Douglas R. Burke and Yo Matsubara. The extent of strength in the club filters. Israel Journal of Mathematics, 114:253–263, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  15. Chen-Chung Chang and H. Jerome Keisler. Model Theory, volume 73 of Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1973.

    MATH  Google Scholar 

  16. James Cummings. Collapsing successors of singulars. Proceedings of the American Mathematical Society, 125(9):2703–2709, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  17. James Cummings. Large cardinal properties of small cardinals. In Set Theory (Curaçao/Barcelona, 1995/1996), pages 23–39. Kluwer Academic, Dordrecht, 1998.

    Google Scholar 

  18. James Cummings, Matthew Foreman, and Menachem Magidor. Canonical structure in the universe of set theory. I. Annals of Pure and Applied Logic, 129(1–3):211–243, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  19. Patrick Dehornoy. Progrès récents sur l’hypothèse du continu (d’après Woodin). Astérisque, 294 147172, 2004.

    Google Scholar 

  20. Oliver Deiser and Dieter Donder. Canonical functions, non-regular ultrafilters and Ulam’s problem on \(\omega\sb1\) . The Journal of Symbolic Logic, 68(3), 2003.

    Google Scholar 

  21. Keith J. Devlin and Håvard Johnsbråten. The Souslin Problem, volume 405 of Lecture Notes in Mathematics. Springer, Berlin, 1974.

    MATH  Google Scholar 

  22. Keith J. Devlin and Saharon Shelah. A weak version of ⋄ which follows from \(2^{\aleph_{0}}<2^{\aleph_{1}}\) . Israel Journal of Mathematics, 29(2–3):239–247, 1978.

    Article  MATH  MathSciNet  Google Scholar 

  23. Dieter Donder. Private communication, 1980.

    Google Scholar 

  24. Hans-Dieter Donder and Jean-Pierre Levinski. Some principles related to Chang’s Conjecture. Annals of Pure and Applied Logic, 45(1):39–101, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  25. Alan Dow, Franklin D. Tall, and William A. R. Weiss. New proofs of the consistency of the normal Moore space conjecture. I. Topology and Its Applications, 37(1):33–51, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  26. Paul C. Eklof and Alan H. Mekler. Almost Free Modules. North-Holland, Amsterdam, 2002. Revised edition.

    MATH  Google Scholar 

  27. Paul Erdős and András Hajnal. On a problem of B. Jónsson. Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, 14:19–23, 1966.

    Google Scholar 

  28. William G. Fleissner. If all normal Moore spaces are metrizable, then there is an inner model with a measurable cardinal. Transactions of the American Mathematical Society, 273(1):365–373, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  29. William G. Fleissner. Normal nonmetrizable Moore space from Continuum Hypothesis or nonexistence of inner models with measurable cardinals. Proceedings of the National Academy of Sciences USA, 79(4):1371–1372, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  30. Matthew Foreman. Calculating quotient algebras for generic embeddings. Israel Journal of Mathematics. To appear.

    Google Scholar 

  31. Matthew Foreman. Forbidden intervals. The Journal of Symbolic Logic, 74(4):1081–1099, 2009.

    Article  MathSciNet  Google Scholar 

  32. Matthew Foreman. Smoke and mirrors: combinatorial properties of small cardinals equiconsistent with huge cardinals. Advances in Mathematics, 222(2):565–595, 2009.

    Article  MATH  MathSciNet  Google Scholar 

  33. Matthew Foreman. Large cardinals and strong model theoretic transfer properties. Transactions of the American Mathematical Society, 272(2):427–463, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  34. Matthew Foreman. More saturated ideals. In Cabal Seminar ’79–’81, volume 1019 of Lecture Notes in Mathematics, pages 1–27. Springer, Berlin, 1983.

    Chapter  Google Scholar 

  35. Matthew Foreman. Potent axioms. Transactions of the American Mathematical Society, 294(1):1–28, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  36. Matthew Foreman. An ℵ 1-dense ideal on ℵ 2. Israel Journal of Mathematics, 108:253–290, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  37. Matthew Foreman. Stationary sets, Chang’s Conjecture and partition theory. In Set Theory (Piscataway, 1999), volume 58 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 73–94. American Mathematical Society, Providence, 2002.

    Google Scholar 

  38. Matthew Foreman. Has the Continuum Hypothesis been settled? In Logic Colloquium ’03 (Helsinki, 2003), volume 24 of Lecture Notes in Logic, pages 56–75. AK Peters, Wellesley, 2006.

    Google Scholar 

  39. Matthew Foreman and András Hajnal. A partition relation for successors of large cardinals. Mathematische Annalen, 325(3):583–623, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  40. Matthew Foreman and Péter Komjáth. The club guessing ideal: commentary on a theorem of Gitik and Shelah. Journal of Mathematical Logic, 5(1):99–147, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  41. Matthew Foreman and Richard Laver. Some downwards transfer properties for ℵ 2. Advances in Mathematics, 67(2):230–238, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  42. Matthew Foreman and Menachem Magidor. Large cardinals and definable counterexamples to the Continuum Hypothesis. Annals of Pure and Applied Logic, 76(1):47–97, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  43. Matthew Foreman and Menachem Magidor. A very weak square principle. The Journal of Symbolic Logic, 62(1):175–196, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  44. Matthew Foreman and Menachem Magidor. Mutually stationary sequences of sets and the non-saturation of the non-stationary ideal on P ϰ (λ). Acta Mathematica, 186(2):271–300, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  45. Matthew Foreman and Stevo Todorcevic. A new Löwenheim-Skolem theorem. Transactions of the American Mathematical Society, 357(5):1693–1715, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  46. Matthew Foreman and W. Hugh Woodin. The Generalized Continuum Hypothesis can fail everywhere. Annals of Mathematics (2), 133(1):1–35, 1991.

    Article  MathSciNet  Google Scholar 

  47. Matthew Foreman, Menachem Magidor, and Saharon Shelah. Martin’s Maximum, saturated ideals, and nonregular ultrafilters. I. Annals of Mathematics (2), 127(1):1–47, 1988.

    Article  MathSciNet  Google Scholar 

  48. Matthew Foreman, Menachem Magidor, and Saharon Shelah. Martin’s maximum, saturated ideals and nonregular ultrafilters. II. Annals of Mathematics (2), 127(3):521–545, 1988.

    Article  MathSciNet  Google Scholar 

  49. Haim Gaifman. Elementary embeddings of models of set theory and certain subtheories. In Axiomatic Set Theory, volume 13(2) of Proceedings of Symposia in Pure Mathematics, pages 33–101. American Mathematical Society, Providence, 1974.

    Google Scholar 

  50. Fred Galvin, Thomas J. Jech, and Menachem Magidor. An ideal game. The Journal of Symbolic Logic, 43(2):284–292, 1978.

    Article  MATH  MathSciNet  Google Scholar 

  51. Moti Gitik. The nonstationary ideal on ℵ 2. Israel Journal of Mathematics, 48(4):257–288, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  52. Moti Gitik. Nonsplitting subset of P κ κ +. The Journal of Symbolic Logic, 50(4):881–894, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  53. Moti Gitik. Some results on the nonstationary ideal. Israel Journal of Mathematics, 92(1–3):61–112, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  54. Moti Gitik. Some results on the nonstationary ideal. II. Israel Journal of Mathematics, 99:175–188, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  55. Moti Gitik and Saharon Shelah. Forcings with ideals and simple forcing notions. Israel Journal of Mathematics, 68(2):129–160, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  56. Moti Gitik and Saharon Shelah. Less saturated ideals. Proceedings of the American Mathematical Society, 125(5):1523–1530, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  57. Kurt Gödel. What is Cantor’s continuum problem? The American Mathematical Monthly, 54:515–525, 1947.

    Article  MathSciNet  Google Scholar 

  58. Noa Goldring. Woodin cardinals and presaturated ideals. Annals of Pure and Applied Logic, 55(3):285–303, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  59. Tetsuya Ishiu. Club guessing sequences and filters. The Journal of Symbolic Logic, 70(4):1037–1071, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  60. Tetsuya Ishiu. A tail club guessing ideal can be saturated without being a restriction of the nonstationary ideal. Notre Dame Journal of Formal Logic, 46(3):327–333, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  61. Thomas J. Jech. Some combinatorial problems concerning uncountable cardinals. Annals of Mathematical Logic, 5:165–198, 1972/1973.

    Article  MathSciNet  Google Scholar 

  62. Thomas J. Jech. Precipitous ideals. In Logic Colloquium ’76 (Oxford, 1976), volume 87 of Studies in Logic and the Foundations of Mathematics, pages 521–530. North-Holland, Amsterdam, 1977.

    Google Scholar 

  63. Thomas J. Jech. Set Theory. Springer Monographs in Mathematics. Springer, Berlin, 2002. The third millennium edition, revised and expanded.

    Google Scholar 

  64. Thomas J. Jech and Karel L. Prikry. On ideals of sets and the power set operation. Bulletin of the American Mathematical Society, 82:593–596, 1976.

    Article  MATH  MathSciNet  Google Scholar 

  65. Thomas J. Jech, Menachem Magidor, William J. Mitchell, and Karel L. Prikry. Precipitous ideals. The Journal of Symbolic Logic, 45(1):1–8, 1980.

    Article  MATH  MathSciNet  Google Scholar 

  66. Yuzuru Kakuda. On a condition for Cohen extensions which preserve precipitous ideals. The Journal of Symbolic Logic, 46(2):296–300, 1981.

    Article  MATH  MathSciNet  Google Scholar 

  67. Akihiro Kanamori. Finest partitions for ultrafilters. The Journal of Symbolic Logic, 51(2):327–332, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  68. Akihiro Kanamori. Partition relations for successor cardinals. Advances in Mathematics, 59(2):152–169, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  69. Akihiro Kanamori. The Higher Infinite. Springer Monographs in Mathematics, Springer, Berlin, 2003. Second edition.

    MATH  Google Scholar 

  70. Akihiro Kanamori and Menachem Magidor. The evolution of large cardinal axioms in set theory. In Higher Set Theory (Oerwolfach, 1977), volume 669 of Lecture Notes in Mathematics, pages 99–275. Springer, Berlin, 1978.

    Chapter  Google Scholar 

  71. Akihiro Kanamori and Alan D. Taylor. Separating ultrafilters on uncountable cardinals. Israel Journal of Mathematics, 47(2–3):131–138, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  72. H. Jerome Keisler. A survey of ultraproducts. In Logic, Methodology and Philosophy of Science (Proceedings of the 1964 International Congress), pages 112–126. North-Holland, Amsterdam, 1965.

    Google Scholar 

  73. Péter Komjáth. On second-category sets. Proceedings of the American Mathematical Society, 107(3):653–654, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  74. Adam Krawczyk and Andrzej Pelc. On families of σ-complete ideals. Fundamenta Mathematicae, 109(2):155–161, 1980.

    MATH  MathSciNet  Google Scholar 

  75. John Krueger. Destroying stationary sets. Israel Journal of Mathematics, 147:285–328, 2005.

    Article  MATH  MathSciNet  Google Scholar 

  76. David W. Kueker. Countable approximations and Löwenheim-Skolem theorems. Annals of Pure and Applied Logic, 11(1):57–103, 1977.

    MATH  MathSciNet  Google Scholar 

  77. Kenneth Kunen. Some applications of iterated ultrapowers in set theory. Annals of Pure and Applied Logic, 1:179–227, 1970.

    MATH  MathSciNet  Google Scholar 

  78. Kenneth Kunen. Elementary embeddings and infinitary combinatorics. The Journal of Symbolic Logic, 36:407–413, 1971.

    Article  MATH  MathSciNet  Google Scholar 

  79. Kenneth Kunen. Saturated ideals. The Journal of Symbolic Logic, 43(1):65–76, 1978.

    Article  MATH  MathSciNet  Google Scholar 

  80. Paul B. Larson. The Stationary Tower. Notes on a Course by W. Hugh Woodin, volume 32 of University Lecture Series. American Mathematical Society, Providence, 2004.

    MATH  Google Scholar 

  81. Paul B. Larson and Saharon Shelah. Bounding by canonical functions, with CH. Journal of Mathematical Logic, 3(2):193–215, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  82. Richard Laver. Making the supercompactness of κ indestructible under κ-directed closed forcing. Israel Journal of Mathematics, 29(4):385–388, 1978.

    Article  MATH  MathSciNet  Google Scholar 

  83. Richard Laver. An (ℵ 2, ℵ 2, ℵ 0)-saturated ideal on ω 1. In Logic Colloquium ’80 (Prague, 1980), volume 108 of Studies in Logic and the Foundations of Mathematics, pages 173–180. North-Holland, Amsterdam, 1982.

    Google Scholar 

  84. Richard Laver. Saturated ideals and nonregular ultrafilters. In Patras Logic Symposion (Patras, 1980), volume 109 of Studies in Logic and the Foundations of Mathematics, pages 297–305. North-Holland, Amsterdam, 1982.

    Chapter  Google Scholar 

  85. Richard Laver. Precipitousness in forcing extensions. Israel Journal of Mathematics, 48(2–3):97–108, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  86. Jean-Pierre Levinski, Menachem Magidor, and Saharon Shelah. Chang’s Conjecture for ℵ ω . Israel Journal of Mathematics, 69(2):161–172, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  87. Kecheng Liu and Saharon Shelah. Cofinalities of elementary substructures of structures on ℵ ω . Israel Journal of Mathematics, 99:189–205, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  88. Menachem Magidor. On the singular cardinals problem. I. Israel Journal of Mathematics, 28(1–2):1–31, 1977.

    Article  MATH  MathSciNet  Google Scholar 

  89. Menachem Magidor. On the existence of nonregular ultrafilters and the cardinality of ultrapowers. Transactions of the American Mathematical Society, 249(1):97–111, 1979.

    Article  MATH  MathSciNet  Google Scholar 

  90. Menachem Magidor. Precipitous ideals and \(\boldsymbol{\Sigma}_{4}^{1}\) sets. Israel Journal of Mathematics, 35(1–2):109–134, 1980.

    Article  MATH  MathSciNet  Google Scholar 

  91. Menachem Magidor. Reflecting stationary sets. The Journal of Symbolic Logic, 47(4):755–771, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  92. Donald A. Martin. Mathematical evidence. In Truth in Mathematics (Mussomeli, 1995), pages 215–231. Oxford University Press, New York, 1998.

    Google Scholar 

  93. Yo Matsubara. Stationary preserving ideals over ℘ κ λ. Journal of the Mathematical Society of Japan, 55(3):827–835, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  94. Yo Matsubara and Saharon Shelah. Nowhere precipitousness of the non-stationary ideal over ℘ κ λ. Journal of Mathematical Logic, 2(1):81–89, 2002.

    Article  MATH  MathSciNet  Google Scholar 

  95. Yo Matsubara and Masahiro Shioya. Nowhere precipitousness of some ideals. The Journal of Symbolic Logic, 63(3):1003–1006, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  96. William J. Mitchell. Aronszajn trees and the independence of the transfer property. Annals of Pure and Applied Logic, 5:21–46, 1972/1973.

    Google Scholar 

  97. Robert L. Moore. Foundations of Point Set Topology, volume 13 of Colloquium Publications. American Mathematical Society, Providence, 1962. Revised edition.

    Google Scholar 

  98. Peter J. Nyikos. A provisional solution to the normal Moore space problem. Proceedings of the American Mathematical Society, 78(3):429–435, 1980.

    Article  MATH  MathSciNet  Google Scholar 

  99. Ralf Schindler. On a Chang conjecture. Israel Journal of Mathematics, 99:221–230, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  100. Ralf Schindler. On a Chang conjecture. II. Archive for Mathematical Logic, 37(4):215–220, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  101. Saharon Shelah. Models with second order properties. III. Omitting types for L(Q). Archiv für Mathematische Logik und Grundlagenforschung, 21(1–2):1–11, 1981.

    Article  MATH  Google Scholar 

  102. Saharon Shelah. Around Classification Theory of Models, volume 1182 of Lecture Notes in Mathematics. Springer, Berlin, 1982.

    Google Scholar 

  103. Saharon Shelah. Proper Forcing, volume 940 of Lecture Notes in Mathematics. Springer, Berlin, 1982.

    MATH  Google Scholar 

  104. Saharon Shelah. Iterated forcing and normal ideals on ω 1. Israel Journal of Mathematics, 60(3):345–380, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  105. Saharon Shelah. Cardinal Arithmetic, volume 29 of Oxford Logic Guides. Clarendon, Oxford, 1994.

    MATH  Google Scholar 

  106. Saharon Shelah. Non-structure theorems. Shelah Website. To appear.

    Google Scholar 

  107. Masahiro Shioya. A saturated stationary subset of P κ κ +. Mathematical Research Letters, 10:493–500, 2003.

    MATH  MathSciNet  Google Scholar 

  108. Roman Sikorski. Boolean Algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 25. Springer, New York, 1969. Third edition.

    MATH  Google Scholar 

  109. Jack Silver. On the singular cardinals problem. In Proceedings of the International Congress of Mathematicians (Vancouver, 1974), Vol. 1, pages 265–268. Canadian Mathematical Congress, Montreal, 1975.

    Google Scholar 

  110. Robert M. Solovay. A model of set theory in which every set of reals is Lebesgue measurable. Annals of Mathematics (2), 92:1–56, 1970.

    Article  MathSciNet  Google Scholar 

  111. Robert M. Solovay. Real-valued measurable cardinals. In Axiomatic Set Theory, volume 13(1) of Proceedings of Symposia in Pure Mathematics, pages 397–428. American Mathematical Society, Providence, 1971.

    Google Scholar 

  112. Robert M. Solovay. Strongly compact cardinals and the GCH. In Proceedings of the Tarski Symposium, volume 25 of Proceedings of Symposia in Pure Mathematics, pages 365–372. American Mathematical Society, Providence, 1974.

    Google Scholar 

  113. Robert M. Solovay, William N. Reinhardt, and Akihiro Kanamori. Strong axioms of infinity and elementary embeddings. Annals of Mathematical Logic, 13(1):73–116, 1978.

    Article  MATH  MathSciNet  Google Scholar 

  114. John R. Steel. The Core Model Iterability Problem, volume 8 of Lecture Notes in Logic. Springer, Berlin, 1996.

    MATH  Google Scholar 

  115. John R. Steel and Robert A. Van Wesep. Two consequences of determinacy consistent with choice. Transactions of the American Mathematical Society, 272(1):67–85, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  116. Franklin D. Tall. Set-theoretic consistency results and topological theorems concerning the normal Moore space conjecture and related problems. Dissertationes Mathematicae (Rozprawy Matematyczne), 148:53, 1977.

    MathSciNet  Google Scholar 

  117. Alfred Tarski. Idealle in vollständige mengenkörpen II. Fundamenta Mathematicae, 33:51–65, 1945.

    MATH  MathSciNet  Google Scholar 

  118. Alfred Tarski. Some problems and results relevant to the foundations of set theory. In Logic, Methodology and Philosophy of Science (Proceedings of the 1960 International Congress, Stanford), pages 125–135. Stanford University Press, Stanford, 1962.

    Google Scholar 

  119. Alan D. Taylor. Regularity properties of ideals and ultrafilters. Annals of Mathematical Logic, 16(1):33–55, 1979.

    Article  MATH  MathSciNet  Google Scholar 

  120. Alan D. Taylor. On saturated sets of ideals and Ulam’s problem. Fundamenta Mathematicae, 109(1):37–53, 1980.

    Google Scholar 

  121. Stevo Todorcevic. A saturated ideal. Handwritten notes, 1987.

    Google Scholar 

  122. Jan Tryba. On Jónsson cardinals with uncountable cofinality. Israel Journal of Mathematics, 49(4):315–324, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  123. Stanisław Ulam. On measure theory in general set theory (doctoral dissertation). Roczniki Polskiego Towarzystwa Matematycznego. Seria II. Wiadomości Matematyczne, 33:155–168, 1997. Reprint of the 1933 original.

    MATH  MathSciNet  Google Scholar 

  124. W. Hugh Woodin. Unpublished work. Oral presentation, ongoing.

    Google Scholar 

  125. W. Hugh Woodin. Supercompact cardinals, sets of reals, and weakly homogeneous trees. Proceedings of the National Academy of Sciences USA, 85:6587–6591, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  126. W. Hugh Woodin. The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal, volume 1 of de Gruyter Series in Logic and Its Applications. de Gruyter, Berlin, 1999.

    MATH  Google Scholar 

  127. W. Hugh Woodin. The Continuum Hypothesis. I. Notices of the American Mathematical Society, 48(6):567–576, 2001.

    MATH  MathSciNet  Google Scholar 

  128. W. Hugh Woodin. The Continuum Hypothesis. II. Notices of the American Mathematical Society, 48(7):681–690, 2001.

    MathSciNet  Google Scholar 

  129. W. Hugh Woodin. The Continuum Hypothesis. In Logic Colloquium 2000, volume 19 of Lecture Notes in Logic, pages 143–197. Association of Symbolic Logic, Urbana, 2005.

    Google Scholar 

  130. Jindřich Zapletal. A new proof of Kunen’s inconsistency. Proceedings of the American Mathematical Society, 124(7):2203–2204, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  131. Stuart Zoble. Stationary reflection and the determinacy of inductive games. PhD thesis, University of California, Berkeley, 2000.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California at Irvine, Irvine, CA, 92697, USA

    Matthew Foreman

Authors
  1. Matthew Foreman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matthew Foreman .

Editor information

Editors and Affiliations

  1. School of Physical Sciences, University of California, Irvine, Irvine, 92697-3875, U.S.A.

    Matthew Foreman

  2. Dept. Manufacturing Engineering, Boston University, St. Mary's St. 15, Boston, 02215, U.S.A.

    Akihiro Kanamori

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer Science+Business Media B.V.

About this chapter

Cite this chapter

Foreman, M. (2010). Ideals and Generic Elementary Embeddings. In: Foreman, M., Kanamori, A. (eds) Handbook of Set Theory. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-5764-9_14

Download citation

Publish with us

Policies and ethics

Search

Navigation