## Arthur W. ApterDistinguished Professor of Mathematics Baruch College of CUNYDistinguished Professor of Mathematics The CUNY Graduate Center

#### Research Interests:

Joel Hamkins, whom I thank for helping me to set up this web page, has created a page containing links of interest to logicians both in the greater New York area and elsewhere. Click here to access these links. Click here to find out information about MAMLS.

#### Slides:

Slides from my lecture "Some Results Concerning Strong Compactness and Supercompactness", which I presented at the Winter Meeting of the ASL held January 17-18, 2003 in Baltimore, can be found by clicking here for the .dvi file, and here for the LaTeX file.

Slides from my lecture "Indestructibility and Strong Compactness", which I presented at Logic Colloquium 2003 held August 14-20, 2003 in Helsinki, Finland can be found by clicking here for the .dvi file, and here for the LaTeX file.

Slides from the lecture I presented at the Baumgartner 60th Birthday Conference, held October 4-5, 2003 at Dartmouth College can be found by clicking here for the .dvi file, and here for the LaTeX file.

#### Current Publication List (last revised 12/27/14):

1. "On the Least Strongly Compact Cardinal", Israel Journal of Mathematics 35, 1980, 225-233.

2. "Changing Cofinalities and Infinite Exponents", Journal of Symbolic Logic 46, 1981, 89-95.

3. "Measurability and Degrees of Strong Compactness", Journal of Symbolic Logic 46, 1981, 180-185.

4. "On a Problem of Silver", Fundamenta Mathematicae 116, 1983, 33-38.

5. "Some Results on Consecutive Large Cardinals", Annals of Pure and Applied Logic 25, 1983, 1-17.

6. "A Generalized Version of the Singular Cardinals Problem", Fundamenta Mathematicae 121, 1984, 99-116.

7. "Successors of Singular Cardinals and Measurability", Advances in Mathematics 55, 1985, 228-241.

8. "An AD-Like Model", Journal of Symbolic Logic 50, 1985, 531-543.

9. "A Cardinal Structure Theorem for an Ultrapower", Canadian Mathematical Bulletin 28, 1985, 472-473.

10. "Some Results on Consecutive Large Cardinals II: Applications of Radin Forcing", Israel Journal of Mathematics 52, 1985, 273-292.

11. (with J. Henle) "Large Cardinal Structures Below $\aleph_\omega$", Journal of Symbolic Logic 51, 1986, 591-603.

12. "On a Problem Inspired by Determinacy", Israel Journal of Mathematics 61, 1988, 256-270.

13. (with M. Gitik) "Some Results on Specker's Problem", Pacific Journal of Mathematics 134, 1988, 227-249.

14. (with C. DiPrisco, J. Henle, and W. Zwicker) "Filter Spaces: Towards a Unified Theory of Large Cardinal and Embedding Axioms", Annals of Pure and Applied Logic 41, 1989, 93-106.

15. "Successors of Singular Cardinals and Measurability Revisited", Journal of Symbolic Logic 55, 1990, 492-501.

16. (with C. DiPrisco, J. Henle, and W. Zwicker) "Filter Spaces II: Limit Ultraproducts and Iterated Embeddings", Acta Cientifica Venezolano 40, 1990, 311-318.

17. "A Note on Strong Compactness and Supercompactness", Bulletin of the London Mathematical Society 23, 1991, 113-115.

18. (with J. Henle) "Relative Consistency Results via Strong Compactness", Fundamenta Mathematicae 139, 1991, 133-149.

19. "Some New Upper Bounds in Consistency Strength for Certain Choiceless Large Cardinal Patterns", Archive for Mathematical Logic 31, 1992, 201-205. .dvi file, Plain TeX file.

20. (with J. Henle) "On Box, Weak Box, and Strong Compactness", Bulletin of the London Mathematical Society 24, 1992, 513-518. .dvi file, Plain TeX file.

21. "On the Class of Measurable Cardinals Without the Axiom of Choice", Israel Journal of Mathematics 79, 1992, 367-379. .dvi file, Plain TeX file.

22. "On the First n Strongly Compact Cardinals", Proceedings of the American Mathematical Society 123, 1995, 2229-2235. .dvi file, Plain TeX file.

23. (with M. Magidor) "Instances of Dependent Choice and the Measurability of $\aleph_{\omega + 1}$", Annals of Pure and Applied Logic 74, 1995, 203-219. .dvi file, Plain TeX file.

24. "AD and Patterns of Singular Cardinals below $\Theta$", Journal of Symbolic Logic 61, 1996, 225-235. .dvi file, Plain TeX file.

25. "A Cardinal Pattern Inspired by AD", Mathematical Logic Quarterly 42, 1996, 211-218. .dvi file, Plain TeX file.

26. (with S. Shelah) "On the Strong Equality between Supercompactness and Strong Compactness", Transactions of the American Mathematical Society 349, 1997, 103-128. .dvi file, Plain TeX file.

27. (with S. Shelah) "Menas' Result is Best Possible", Transactions of the American Mathematical Society 349, 1997, 2007-2034. .dvi file, Plain TeX file.

28. "More on the Least Strongly Compact Cardinal", Mathematical Logic Quarterly 43, 1997, 427-430. .dvi file, Plain TeX file.

29. "Patterns of Compact Cardinals", Annals of Pure and Applied Logic 89, 1997, 101-115. .dvi file, Plain TeX file.

30. "Laver Indestructibility and the Class of Compact Cardinals", Journal of Symbolic Logic 63, 1998, 149-157. .dvi file, Plain TeX file.

31. (with M. Gitik) "The Least Measurable can be Strongly Compact and Indestructible", Journal of Symbolic Logic 63, 1998, 1404-1412. .dvi file, Plain TeX file.

32. "Forcing the Least Measurable to Violate GCH", Mathematical Logic Quarterly 45, 1999, 551-560. .dvi file, LaTeX file.

33. "On Measurable Limits of Compact Cardinals", Journal of Symbolic Logic 64, 1999, 1675-1688. .dvi file, Plain TeX file.

34. (with J. D. Hamkins) "Universal Indestructibility", Kobe Journal of Mathematics 16, 1999, 119-130. .dvi file, Plain TeX file.

35. (with J. Henle and S. Jackson) "The Calculus of Partition Sequences, Changing Cofinalities, and a Question of Woodin", Transactions of the American Mathematical Society 352, 2000, 969-1003. .dvi file, LaTeX file.

36. (with J. Cummings) "A Global Version of a Theorem of Ben-David and Magidor", Annals of Pure and Applied Logic 102, 2000, 199-222. .dvi file, LaTeX file.

37. "A New Proof of a Theorem of Magidor", Archive for Mathematical Logic 39, 2000, 209-211. .dvi file, Plain TeX file.

38. "On a Problem of Woodin", Archive for Mathematical Logic 39, 2000, 253-259. .dvi file, Plain TeX file.

39. "Strong Compactness and a Global Version of a Theorem of Ben-David and Magidor", Mathematical Logic Quarterly 46, 2000, 453-459. .dvi file, LaTeX file.

40. (with J. Cummings) "Identity Crises and Strong Compactness", Journal of Symbolic Logic 65, 2000, 1895-1910. .dvi file, Plain TeX file.

41. "A Note on Strong Compactness and Resurrectibility", Fundamenta Mathematicae 165, 2000, 285-290. .dvi file, LaTeX file.

42. (with J. Cummings) "Identity Crises and Strong Compactness II: Strong Cardinals", Archive for Mathematical Logic 40, 2001, 25-38. .dvi file.

43. "Some Remarks on Normal Measures and Measurable Cardinals", Mathematical Logic Quarterly 47, 2001, 35-44. .dvi file, LaTeX file.

44. "Strong Compactness, Measurability, and the Class of Supercompact Cardinals", Fundamenta Mathematicae 167, 2001, 65-78. .dvi file, LaTeX file.

45. "Supercompactness and Measurable Limits of Strong Cardinals", Journal of Symbolic Logic 66, 2001, 629-639. .dvi file, LaTeX file.

46. "On the Consistency Strength of Two Choiceless Cardinal Patterns", Notre Dame Journal of Formal Logic 40, 1999, 341-345. Note: This paper actually appeared during the summer of 2001. .dvi file, LaTeX file.

47. (with M. Dzamonja) "Some Remarks on a Question of D. H. Fremlin Regarding $\epsilon$-Density", Archive for Mathematical Logic 40, 2001, 531-540. .dvi file, LaTeX file.

48. (with J. D. Hamkins) "Indestructible Weakly Compact Cardinals and the Necessity of Supercompactness for Certain Proof Schemata", Mathematical Logic Quarterly 47, 2001, 563-571. .dvi file, LaTeX file.

49. "Expanding $\kappa$'s Power Set in its Ultrapowers", Radovi Matematicki 10, 2001, 149-156. .dvi file, LaTeX file.

50. "Some Structural Results Concerning Supercompact Cardinals", Journal of Symbolic Logic 66, 2001, 1919-1927. .dvi file, LaTeX file.

51. "On Level by Level Equivalence and Inequivalence between Strong Compactness and Supercompactness", Fundamenta Mathematicae 171, 2002, 77-92. .dvi file, LaTeX file.

52. (with J. D. Hamkins) "Indestructibility and the Level-by-Level Agreement between Strong Compactness and Supercompactness", Journal of Symbolic Logic 67, 2002, 820-840. .dvi file.

53. (with J. Cummings) "Blowing up the Power Set of the Least Measurable", Journal of Symbolic Logic 67, 2002, 915-923. .dvi file.

54. "On the Non-Extendibility of Strongness and Supercompactness through Strong Compactness", Fundamenta Mathematicae 174, 2002, 87-96. .dvi file.

55. "Strong Cardinals can be Fully Laver Indestructible", Mathematical Logic Quarterly 48, 2002, 499-507. .dvi file, LaTeX file.

56. "Aspects of Strong Compactness, Measurability, and Indestructibility", Archive for Mathematical Logic 41, 2002, 705-719. .dvi file, LaTeX file.

57. "On the Level by Level Equivalence between Strong Compactness and Strongness", Journal of the Mathematical Society of Japan 55, 2003, 47-58. .dvi file.

58. (with J. D. Hamkins) "Exactly Controlling the Non-Supercompact Strongly Compact Cardinals", Journal of Symbolic Logic 68, 2003, 669-688. .dvi file, LaTeX file.

59. "Characterizing Strong Compactness via Strongness", Mathematical Logic Quarterly 49, 2003, 375-384. .dvi file, LaTeX file.

60. "Failures of GCH and the Level by Level Equivalence between Strong Compactness and Supercompactness", Mathematical Logic Quarterly 49, 2003, 587-597. .dvi file, LaTeX file.

61. "Indestructibility, Strongness, and Level by Level Equivalence", Fundamenta Mathematicae 177, 2003, 45-54. .dvi file, LaTeX file.

62. "Some Remarks on Indestructibility and Hamkins' Lottery Preparation", Archive for Mathematical Logic 42, 2003, 717-735. .dvi file, LaTeX file.

63. "Level by Level Equivalence and Strong Compactness", Mathematical Logic Quarterly 50, 2004, 51-64. .dvi file, LaTeX file.

64. "Supercompactness and Partial Level by Level Equivalence between Strong Compactness and Strongness", Fundamenta Mathematicae 182, 2004, 123-136. .dvi file, LaTeX file.

65. (with G. Sargsyan) "Jonsson-like Partition Relations and j : V ---> V", Journal of Symbolic Logic 69, 2004, 1267-1281. .dvi file, LaTeX file.

66. "Removing Laver Functions from Supercompactness Arguments", Mathematical Logic Quarterly 51, 2005, 154-156. .dvi file, LaTeX file.

67. "An Easton Theorem for Level by Level Equivalence", Mathematical Logic Quarterly 51, 2005, 247-253. .dvi file, LaTeX file.

68. "Diamond, Square, and Level by Level Equivalence", Archive for Mathematical Logic 44, 2005, 387-395. .dvi file, LaTeX file.

69. "On a Problem of Foreman and Magidor", Archive for Mathematical Logic 44, 2005, 493-498. .dvi file, LaTeX file.

70. (with G. Sargsyan) "Can A Large Cardinal Be Forced From A Condition Implying Its Negation?", Proceedings of the American Mathematical Society 133, 2005, 3103-3108. .dvi file, LaTeX file.

71. "Universal Partial Indestructibility and Strong Compactness", Mathematical Logic Quarterly 51, 2005, 524-531. .dvi file, LaTeX file.

72. "Universal Indestructibility is Consistent with Two Strongly Compact Cardinals", Bulletin of the Polish Academy of Sciences (Mathematics) 53, 2005, 131-135. .dvi file, LaTeX file.

73. (with G. Sargsyan) "Identity Crises and Strong Compactness III: Woodin Cardinals", Archive for Mathematical Logic 45, 2006, 307-322. .dvi file, LaTeX file.

74. "Indestructibility and Strong Compactness", Proceedings of Logic Colloquium 2003, Lecture Notes in Logic 24, 2006, 27-37. .dvi file, LaTeX file.

75. "How Many Normal Measures Can $\aleph_{\omega + 1}$ Carry?", Fundamenta Mathematicae 191, 2006, 57-66. .dvi file, LaTeX file.

76. "Supercompactness and Measurable Limits of Strong Cardinals II: Applications to Level by Level Equivalence", Mathematical Logic Quarterly 52, 2006, 457-463. .dvi file, LaTeX file.

77. (with P. Koepke) "The Consistency Strength of $\aleph_\omega$ and $\aleph_{\omega_1}$ being Rowbottom Cardinals without the Axiom of Choice", Archive for Mathematical Logic 45, 2006, 721-737. .dvi file, LaTeX file.

78. "Failures of SCH and Level by Level Equivalence", Archive for Mathematical Logic 45, 2006, 831-838. .dvi file, LaTeX file.

79. "The Least Strongly Compact can be the Least Strong and Indestructible", Annals of Pure and Applied Logic 144, 2006, 33-42. .dvi file, LaTeX file.

80. "Indestructibility and Level by Level Equivalence and Inequivalence", Mathematical Logic Quarterly 53, 2007, 78-85. .dvi file, LaTeX file.

81. "Supercompactness and Level by Level Equivalence are Compatible with Indestructibility for Strong Compactness", Archive for Mathematical Logic 46, 2007, 155-163. .dvi file, LaTeX file.

82. "Level by Level Equivalence and the Number of Normal Measures over $P_\kappa(\lambda)$", Fundamenta Mathematicae 194, 2007, 253-265. .dvi file, LaTeX file.

83. (with G. Sargsyan) "A Reduction in Consistency Strength for Universal Indestructibility", Bulletin of the Polish Academy of Sciences (Mathematics) 55, 2007, 1-6. .dvi file, LaTeX file.

84. (with J. Cummings and J. D. Hamkins) "Large Cardinals with Few Measures", Proceedings of the American Mathematical Society 135, 2007, 2291-2300. .dvi file, LaTeX file.

85. "Reducing the Consistency Strength of an Indestructibility Theorem", Mathematical Logic Quarterly 54, 2008, 288-293. .dvi file, LaTeX file.

86. (with J. Cummings) "An L-like Model Containing Very Large Cardinals", Archive for Mathematical Logic 47, 2008, 65-78. .dvi file, LaTeX file.

87. "Indestructibility and Measurable Cardinals with Few and Many Measures", Archive for Mathematical Logic 47, 2008, 101-110. .dvi file, LaTeX file.

88. (with G. Sargsyan) "Universal Indestructibility for Degrees of Supercompactness and Strongly Compact Cardinals", Archive for Mathematical Logic 47, 2008, 133-142. .dvi file, LaTeX file.

89. (with P. Koepke) "Making All Cardinals Almost Ramsey", Archive for Mathematical Logic 47, 2008, 769-783. .dvi file, LaTeX file.

90. "On the Number of Normal Measures $\aleph_1$ and $\aleph_2$ can Carry", Tbilisi Mathematical Journal 1, 2008, 9-14. Note: This paper may be accessed online at http://ncst.org.ge/Journals/TMJ/ by clicking on the link "Most recent volume". .dvi file, .pdf file, LaTeX file.

91. "A Note on Indestructibility and Strong Compactness", Bulletin of the Polish Academy of Sciences (Mathematics) 56, 2008, 191-197. .dvi file, .pdf file, LaTeX file.

92. "Stationary Reflection and Level by Level Equivalence", Colloquium Mathematicum 115, 2009, 113-128. .dvi file, .pdf file, LaTeX file.

93. "Indestructibility and Stationary Reflection", Mathematical Logic Quarterly 55, 2009, 228-236. .dvi file, .pdf file, LaTeX file.

94. "Indestructibility under Adding Cohen Subsets and Level by Level Equivalence", Mathematical Logic Quarterly 55, 2009, 271-279. .dvi file, .pdf file, LaTeX file.

95. "Indestructibility, Strong Compactness, and Level by Level Equivalence", Fundamenta Mathematicae 204, 2009, 113-126. .dvi file, .pdf file, LaTeX file.

96. "Sandwiching the Consistency Strength of Two Global Choiceless Cardinal Patterns", Bulletin of the Polish Academy of Sciences (Mathematics) 57, 2009, 189-197. .dvi file, .pdf file, LaTeX file.

97. "L-like Combinatorial Principles and Level by Level Equivalence", Bulletin of the Polish Academy of Sciences (Mathematics) 57, 2009, 199-207. .dvi file, .pdf file, LaTeX file.

98. "Tallness and Level by Level Equivalence and Inequivalence", Mathematical Logic Quarterly 56, 2010, 4-12. .dvi file, .pdf file, LaTeX file.

99. (with G. Sargsyan) "An Equiconsistency for Universal Indestructibility", Journal of Symbolic Logic 75, 2010, 314-322. .dvi file, .pdf file, LaTeX file.

100. "How Many Normal Measures Can $\aleph_{\omega_1 + 1}$ Carry?", Mathematical Logic Quarterly 56, 2010, 164-170. .dvi file, .pdf file, LaTeX file.

101. (with P. Koepke) "The Consistency Strength of Choiceless Failures of SCH", Journal of Symbolic Logic 75, 2010, 1066-1080. .dvi file, .pdf file, LaTeX file.

102. "Indestructibility, Instances of Strong Compactness, and Level by Level Inequivalence", Archive for Mathematical Logic 49, 2010, 725-741. .dvi file, .pdf file, LaTeX file.

103. "Indestructibility, HOD, and the Ground Axiom", Mathematical Logic Quarterly 57, 2011, 261-265. .dvi file, .pdf file, LaTeX file.

104. "A Remark on the Tree Property in a Choiceless Context", Archive for Mathematical Logic 50, 2011, 585-590. .dvi file, .pdf file, LaTeX file.

105. (with Sh. Friedman) "Coding into HOD via Normal Measures with Some Applications", Mathematical Logic Quarterly 57, 2011, 366-372. .dvi file, .pdf file, LaTeX file, .bib file.

106. "Level by Level Inequivalence beyond Measurability", Archive for Mathematical Logic 50, 2011, 707-712. .dvi file, .pdf file, LaTeX file.

107. "Indestructibility, Measurability, and Degrees of Supercompactness", Mathematical Logic Quarterly 58, 2012, 75-82. .dvi file, .pdf file, LaTeX file.

108. (with V. Gitman and J. D. Hamkins) "Inner Models with Large Cardinal Features Usually Obtained by Forcing", Archive for Mathematical Logic 51, 2012, 257-283. .dvi file, .pdf file, LaTeX file, .bib file.

109. "Some Applications of Sargsyan's Equiconsistency Method", Fundamenta Mathematicae 216, 2012, 207-222. .dvi file, .pdf file, LaTeX file.

110. "The Wholeness Axioms and the Class of Supercompact Cardinals", Bulletin of the Polish Academy of Sciences (Mathematics) 60, 2012, 101-111. .dvi file, .pdf file, LaTeX file.

111. (with M. Gitik and G. Sargsyan) "Indestructible Strong Compactness but not Supercompactness", Annals of Pure and Applied Logic 163, 2012, 1237-1242. .dvi file, .pdf file, LaTeX file.

112. "Level by Level Inequivalence, Strong Compactness, and GCH", Bulletin of the Polish Academy of Sciences (Mathematics) 60, 2012, 201-209. .dvi file, .pdf file, LaTeX file.

113. "More Easton Theorems for Level by Level Equivalence", Colloquium Mathematicum 128, 2012, 69-86. .dvi file, .pdf file, LaTeX file.

114. "On Some Questions Concerning Strong Compactness", Archive for Mathematical Logic 51, 2012, 819-829. .dvi file, .pdf file, LaTeX file.

115. (with S. Jackson and B. Loewe) "Cofinality and Measurability of the First Three Uncountable Cardinals", Transactions of the American Mathematical Society 365, 2013, 59-98. .dvi file, .pdf file, LaTeX file, .bib file.

116. (with B. Cody) "Consecutive Singular Cardinals and the Continuum Function", Notre Dame Journal of Formal Logic 54, 2013, 125-136. .dvi file, .pdf file, LaTeX file.

117. (with J. Cummings and J. D. Hamkins) "Singular Cardinals and Strong Extenders", Central European Journal of Mathematics 11, 2013, 1628-1634. .dvi file, .pdf file, LaTeX file.

118. "Indestructible Strong Compactness and Level by Level Inequivalence", Mathematical Logic Quarterly 59, 2013, 371-377. .dvi file, .pdf file, LaTeX file.

119. "Some Remarks on Tall Cardinals and Failures of GCH", Bulletin of the Polish Academy of Sciences (Mathematics) 61, 2013, 97-106. .dvi file, .pdf file, LaTeX file.

120. "A Note on Powers of Singular Strong Limit Cardinals", Infinity, Computability, and Metamathematics: Festschrift celebrating the 60th birthdays of Peter Koepke and Philip Welch, College Publications, Volume 23, 2014, 1-3. .dvi file, .pdf file, LaTeX file.

121. "Singular Failures of GCH and Level by Level Equivalence", Bulletin of the Polish Academy of Sciences (Mathematics) 62, 2014, 11-21. .dvi file, .pdf file, LaTeX file.

122. (with M. Gitik) "On Tall Cardinals and Some Related Generalizations", Israel Journal of Mathematics 202, 2014, 343-373. .dvi file, .pdf file, LaTeX file.

123. "Inaccessible Cardinals, Failures of GCH, and Level by Level Equivalence", Notre Dame Journal of Formal Logic 55, 2014, 431-444. .dvi file, .pdf file, LaTeX file.

124. (with P. Koepke and I. Dimitriou) "The First Measurable Cardinal can be the First Uncountable Regular Cardinal at Any Successor Height", Mathematical Logic Quarterly 60, 2014, 471-486. .pdf file, LaTeX file, .bib file.

125. (with Sh. Friedman) "HOD-Supercompactness, Indestructibility, and Level by Level Equivalence", to appear in the Bulletin of the Polish Academy of Sciences (Mathematics). .dvi file, .pdf file, LaTeX file, .bib file.

126. "Indestructibility and Destructible Measurable Cardinals", to appear in the Archive for Mathematical Logic (the special volume in honor of Richard Laver). .dvi file, .pdf file, LaTeX file.

127. "On the Consistency Strength of Level by Level Inequivalence", to appear in the Archive for Mathematical Logic (the special volume in honor of Jim Baumgartner). .dvi file, .pdf file, LaTeX file.

128. "A Universal Indestructibility Theorem Compatible with Level by Level Equivalence", submitted for publication to the Notre Dame Journal of Formal Logic. .dvi file, .pdf file, LaTeX file.

129. (with P. Koepke and I. Dimitriou) "All Uncountable Cardinals in the Gitik Model are Almost Ramsey and Carry Rowbottom Filters", submitted for publication to the Mathematical Logic Quarterly. .pdf file, LaTeX file, .bib file.

130. "Indestructible Strong Compactness and Level by Level Equivalence with No Large Cardinal Restrictions", submitted for publication to the Archive for Mathematical Logic. .dvi file, .pdf file, LaTeX file.

131. "Mixed Levels of Indestructibility", submitted for publication to Bulletin of the Polish Academy of Sciences (Mathematics). .dvi file, .pdf file, LaTeX file.

132. "Indestructibility and the Levinski Property", submitted for publication to the Mathematical Logic Quarterly. .pdf file, LaTeX file.