Climb into Cantor’s Attic, where you will find infinities large and small. We aim to provide a comprehensive resource of information about all notions of mathematical infinity.

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The Cantor's attic library

Step up the ladder towards wisdom, photo by Sigfrid Lundberg

Welcome to the library, our central repository for references cited here on Cantor’s attic.

Library holdings

  1. Abramson, F., Harrington, L., Kleinberg, E., & Zwicker, W. (1977). Flipping properties: a unifying thread in the theory of large cardinals. Ann. Math. Logic, 12(1), 25–58.
  2. Apter, A. W. Some applications of Sargsyan’s equiconsistency method. Fund. Math., 216, 207–222.
  3. Baaz, M., Papadimitriou, C. H., Putnam, H. W., Scott, D. S., & Harper, C. L. (2011). Kurt Gödel and the Foundations of Mathematics: Horizons of Truth. Cambridge University Press.
  4. Bagaria, J. (2006). Axioms of generic absoluteness. Logic Colloquium 2002.
  5. Bagaria, J., & Bosch, R. (2004). Proper forcing extensions and Solovay models. Archive for Mathematical Logic.
  6. Bagaria, J., & Bosch, R. (2007). Generic absoluteness under projective forcing. Fundamenta Mathematicae, 194, 95–120.
  7. Bagaria, J. (2012). \(C^{(n)}\)-cardinals. Archive for Mathematical Logic, 51(3–4), 213–240.
  8. Bagaria, J., Casacuberta, C., Mathias, A. R. D., & Rosický, J. Definable orthogonality classes in accessible categories are small. Journal of the European Mathematical Society, 17(3), 549–589.
  9. Bagaria, J., Hamkins, J. D., Tsaprounis, K., & Usuba, T. (2013). Superstrong and other large cardinals are never Laver indestructible. Archive for Mathematical Logic, 55(1-2), 19–35.
  10. Bagaria, J. (2017). Large Cardinals beyond Choice.
  11. Bagaria, J., Gitman, V., & Schindler, R. (2017). Generic Vopěnkaś Principle, remarkable cardinals, and the weak Proper Forcing Axiom. Arch. Math. Logic, 56(1-2), 1–20.
  12. Baumgartner, J. (1975). Ineffability properties of cardinals. I. In Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vol. I (pp. 109–130. Colloq. Math. Soc. János Bolyai, Vol. 10). North-Holland.
  13. Blass, A. (2010). Chapter 6: Cardinal characteristics of the continuum. Handbook of Set Theory. ablass/hbk.pdf
  14. Blass, A. (1976). Exact functors and measurable cardinals. Pacific J. Math., 63(2), 335–346.
  15. Boney, W. (2017). Model Theoretic Characterizations of Large Cardinals.
  16. Bosch, R. (2006). Small Definably-large Cardinals. Set Theory. Trends in Mathematics, 55–82.
  17. Cantor, G. (1955). Contributions to the Founding of the Theory of Transfinite Numbers (P. Jourdain, Ed.). Dover.
  18. Carmody, E. K. (2015). Force to change large cardinal strength.
  19. Carmody, E., Gitman, V., & Habič, M. E. (2016). A Mitchell-like order for Ramsey and Ramsey-like cardinals.
  20. Chang, C. C. (1971). Sets Constructible Using \(\mathcal {L}_{κ,κ}\). In Axiomatic set theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) (pp. 1–8). Amer. Math. Soc.
  21. Cody, B., Gitik, M., Hamkins, J. D., & Schanker, J. (2013). The Least Weakly Compact Cardinal Can Be Unfoldable, Weakly Measurable and Nearly θ-Supercompact.
  22. Cody, B., & Gitman, V. (2015). Eastonś theorem for Ramsey and strongly Ramsey cardinals. Annals of Pure and Applied Logic, 166(9), 934–952.
  23. Corazza, P. (2000). The Wholeness Axiom and Laver sequences. Annals of Pure and Applied Logic, 157–260.
  24. Corazza, P. (2003). The gap between \({\rm I}_3\)and the wholeness axiom. Fund. Math., 179(1), 43–60.
  25. Corazza, P. (2006). The spectrum of elementary embeddings j : V \to V. Annals of Pure and Applied Logic, 139(1–3), 327–399.
  26. Corazza, P. (2010). The Axiom of Infinity and transformations j: V \to V. Bulletin of Symbolic Logic, 16(1), 37–84.
  27. Daghighi, A. S., & Pourmahdian, M. (2018). On Some Properties of Shelah Cardinals. Bull. Iran. Math. Soc., 44(5), 1117–1124.
  28. Dimonte, V. (2017). I0 and rank-into-rank axioms.
  29. Dimopoulos, S. (2019). Woodin for strong compactness cardinals. The Journal of Symbolic Logic, 84(1), 301–319.
  30. Dodd, A., & Jensen, R. (1981). The core model. Ann. Math. Logic, 20(1), 43–75.
  31. Donder, H.-D., & Koepke, P. (1998). On the Consistency Strength of Áccessible\’Jónsson Cardinals and of the Weak Chang Conjecture. Annals of Pure and Applied Logic.
  32. Donder, H.-D., & Levinski, J.-P. (1989). Some principles related to Changś conjecture. Annals of Pure and Applied Logic.
  33. Drake, F. (1974). Set Theory: An Introduction to Large Cardinals. North-Holland Pub. Co.
  34. Erdős, P., & Hajnal, A. (1962). Some remarks concerning our paper “On the structure of set-mappings\’\’. Non-existence of a two-valued σ-measure for the first uncountable inaccessible cardinal. Acta Math. Acad. Sci. Hungar., 13, 223–226.
  35. Erdős, P., & Hajnal, A. (1958). On the structure of set-mappings. Acta Math. Acad. Sci. Hungar, 9, 111–131.
  36. Eskrew, M., & Hayut, Y. (2016). On the consistency of local and global versions of Changś Conjecture.
  37. Esser, O. (1996). Inconsistency of GPK+AFA. Mathematical Logic Quarterly, 42, 104–108.
  38. Esser, O. (1997). An Interpretation of the Zermelo-Fraenkel Set Theory and the Kelley-Morse Set Theory in a Positive Theory. Mathematical Logic Quarterly, 43, 369–377.
  39. Esser, O. (1999). On the Consistency of a Positive Theory. Mathematical Logic Quarterly, 45, 105–116.
  40. Esser, O. (2000). Inconsistency of the Axiom of Choice with the Positive Theory GPK^+_∞. Journal of Symbolic Logic, 65(4), 1911–1916.
  41. Esser, O. (2003). On the axiom of extensionality in the positive set theory. Mathematical Logic Quarterly, 19, 97–100.
  42. Evans, C. D. A., & Hamkins, J. D. Transfinite game values in infinite chess.
  43. Feng, Q. (1990). A hierarchy of Ramsey cardinals. Annals of Pure and Applied Logic, 49(3), 257–277.
  44. Foreman, M., & Kanamori, A. (2010). Handbook of Set Theory (M. Foreman & A. Kanamori, Eds.; First). Springer.
  45. Forti, M., & Hinnion, R. (1989). The Consistency Problem for Positive Comprehension Principles. J. Symbolic Logic, 54(4), 1401–1418.
  46. Friedman, H. M. (1998). Subtle cardinals and linear orderings.
  47. Fuchs, G., Hamkins, J. D., & Reitz, J. (2015). Set-theoretic geology. Annals of Pure and Applied Logic, 166(4), 464–501.
  48. Gaifman, H. (1974). Elementary embeddings of models of set-theory and certain subtheories. In Axiomatic set theory (Proc. Sympos. Pure Math., Vol. XIII, Part II, Univ. California, Los Angeles, Calif., 1967) (pp. 33–101). Amer. Math. Soc.
  49. Gitman, V. (2011). Ramsey-like cardinals. The Journal of Symbolic Logic, 76(2), 519–540.
  50. Gitman, V., & Welch, P. (2011). Ramsey-like cardinals II. J. Symbolic Logic, 76(2), 541–560.
  51. Gitman, V., & Hamkins, J. D. (2018). A model of the generic Vopěnka principle in which the ordinals are not Mahlo.
  52. Gitman, V., & Johnstone, T. A. Indestructibility for Ramsey and Ramsey-like cardinals.
  53. Gitman, V., & Shindler, R. Virtual large cardinals.
  54. Goldblatt, R. (1998). Lectures on the Hyperreals. Springer.
  55. Goldstern, M., & Shelah, S. (1995). The Bounded Proper Forcing Axiom. J. Symbolic Logic, 60(1), 58–73.
  56. Golshani, M. (2017). An Easton like theorem in the presence of Shelah cardinals. M. Arch. Math. Logic, 56(3-4), 273–287.
  57. Hamkins, J. D., & Lewis, A. (2000). Infinite time Turing machines. J. Symbolic Logic, 65(2), 567–604.
  58. Hamkins, J. D. (2001). The wholeness axioms and V=HOD. Arch. Math. Logic, 40(1), 1–8.
  59. Hamkins, J. D. (2002). Infinite time Turing machines. Minds and Machines, 12(4), 521–539.
  60. Hamkins, J. D. (2004). Supertask computation. Classical and New Paradigms of Computation and Their Complexity Hierarchies, 23, 141–158.
  61. Hamkins, J. D. (2008). Unfoldable cardinals and the GCH.
  62. Hamkins, J. D. (2009). Tall cardinals. MLQ Math. Log. Q., 55(1), 68–86.
  63. Hamkins, J. D., & Johnstone, T. A. (2010). Indestructible strong un-foldability. Notre Dame J. Form. Log., 51(3), 291–321.
  64. Hamkins, J. D., & Johnstone, T. A. (2014). Resurrection axioms and uplifting cardinals.
  65. Hamkins, J. D., & Johnstone, T. A. (2014). Strongly uplifting cardinals and the boldface resurrection axioms.
  66. Hamkins, J. D. (2016). The Vopěnka principle is inequivalent to but conservative over the Vopěnka scheme.
  67. Hauser, K. (1991). Indescribable Cardinals and Elementary Embeddings. 56(2), 439–457.
  68. Holy, P., & Schlicht, P. (2018). A hierarchy of Ramsey-like cardinals. Fundamenta Mathematicae, 242, 49–74.
  69. Jackson, S., Ketchersid, R., Schlutzenberg, F., & Woodin, W. H. (2015). Determinacy and Jónsson cardinals in L(\mathbbR).
  70. Jech, T. J. (2003). Set Theory (Third). Springer-Verlag.
  71. Jensen, R., & Kunen, K. (1969). Some combinatorial properties of L and V. raesch/org/jensen.html
  72. Kanamori, A., & Magidor, M. (1978). The evolution of large cardinal axioms in set theory. In Higher set theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1977) (Vol. 669, pp. 99–275). Springer.
  73. Kanamori, A., Reinhardt, W. N., & Solovay, R. M. (1978). Strong axioms of infinity and elementary embeddings.
  74. Kanamori, A., & Awerbuch-Friedlander, T. (1990). The compleat 0†. Mathematical Logic Quarterly, 36(2), 133–141.
  75. Kanamori, A. (2009). The higher infinite (Second, p. xxii+536). Springer-Verlag.
  76. Kentaro, S. (2007). Double helix in large large cardinals and iteration of elementary embeddings. Annals of Pure and Applied Logic, 146(2-3), 199–236.
  77. Ketonen, J. (1974). Some combinatorial principles. Trans. Amer. Math. Soc., 188, 387–394.
  78. Koellner, P., & Woodin, W. H. (2010). Chapter 23: Large cardinals from Determinacy. Handbook of Set Theory.
  79. Kunen, K. (1978). Saturated Ideals. J. Symbolic Logic, 43(1), 65–76.
  80. Larson, P. B. (2013). A brief history of determinacy.
  81. Laver, R. (1997). Implications between strong large cardinal axioms. Ann. Math. Logic, 90(1–3), 79–90.
  82. Leshem, A. (2000). On the consistency of the definable tree property on \aleph_1. J. Symbolic Logic, 65(3), 1204–1214.
  83. Maddy, P. (1988). Believing the axioms. I. J. Symbolic Logic, 53(2), 181–511.
  84. Maddy, P. (1988). Believing the axioms. II. J. Symbolic Logic, 53(3), 736–764.
  85. Madore, D. (2017). A zoo of ordinals. david/math/ordinal-zoo.pdf
  86. Makowsky, J. (1985). Vopěnkaś Principle and Compact Logics. J. Symbol Logic.
  87. Mitchell, W. J. (1997). Jónsson Cardinals, Erdős Cardinals, and the Core Model. J. Symbol Logic.
  88. Mitchell, W. J. (2001). The Covering Lemma. Handbook of Set Theory. jech/library/mitchell/
  89. Miyamoto, T. (1998). A note on weak segments of PFA. Proceedings of the Sixth Asian Logic Conference, 175–197.
  90. Nielsen, D. S., & Welch, P. (2018). Games and Ramsey-like cardinals.
  91. Perlmutter, N. (2010). The large cardinals between supercompact and almost-huge.
  92. Rathjen, M. (2006). The art of ordinal analysis.
  93. Schanker, J. A. (2012). Partial near supercompactness. Ann. Pure Appl. Logic.
  94. Schanker, J. A. (2011). Weakly measurable cardinals. MLQ Math. Log. Q., 57(3), 266–280.
  95. Schanker, J. A. (2011). Weakly measurable cardinals and partial near supercompactness [PhD thesis]. CUNY Graduate Center.
  96. Schindler, R.-D. (2000). Proper forcing and remarkable cardinals. Bull. Symbolic Logic, 6(2), 176–184.
  97. Sharpe, I., & Welch, P. (2011). Greatly Erdős cardinals with some generalizations to the Chang and Ramsey properties. Ann. Pure Appl. Logic, 162(11), 863–902.
  98. Shelah, S. (1994). Cardinal Arithmetic. Oxford Logic Guides, 29.
  99. Silver, J. (1970). A large cardinal in the constructible universe. Fund. Math., 69, 93–100.
  100. Silver, J. (1971). Some applications of model theory in set theory. Ann. Math. Logic, 3(1), 45–110.
  101. Suzuki, A. (1998). Non-existence of generic elementary embeddings into the ground model. Tsukuba J. Math., 22(2), 343–347.
  102. Suzuki, A. (1999). No elementary embedding from V into V is definable from parameters. J. Symbolic Logic, 64(4), 1591–1594.
  103. Trang, N., & Wilson, T. (2016). Determinacy from Strong Compactness of \omega_1.
  104. Tryba, J. (1983). On Jónsson cardinals with uncountable cofinality. Israel Journal of Mathematics, 49(4).
  105. Usuba, T. (2017). The downward directed grounds hypothesis and very large cardinals. Journal of Mathematical Logic, 17(02), 1750009.
  106. Usuba, T. (2019). Extendible cardinals and the mantle. Archive for Mathematical Logic, 58(1-2), 71–75.
  107. Viale, M., & Weiß, C. (2011). On the consistency strength of the proper forcing axiom. Advances in Mathematics, 228(5), 2672–2687.
  108. Villaveces, A. (1996). Chains of End Elementary Extensions of Models of Set Theory. JSTOR.
  109. Welch, P. (1998). Some remarks on the maximality of Inner Models. Logic Colloquium.
  110. Welch, P. (2000). The Lengths of Infinite Time Turing Machine Computations. Bulletin of the London Mathematical Society, 32(2), 129–136.
  111. Wilson, T. M. (2018). Weakly remarkable cardinals, Erdős cardinals, and the generic Vopěnka principle.
  112. Woodin, W. H. (2010). Suitable extender models I. Journal of Mathematical Logic, 10(01n02), 101–339.
  113. Woodin, W. H. (2011). Suitable extender models II: beyond ω-huge. Journal of Mathematical Logic, 11(02), 115–436.
  114. Zapletal, J. (1996). A new proof of Kunenś inconsistency. Proc. Amer. Math. Soc., 124(7), 2203–2204.

User instructions

Cantor’s attic users may make contributions to the library, in bibtex format, and then cite those references in other articles. Edit the /_bibliography/references.bib file to make your contribution.