Model Theory

Proof Theory and Intuitionism

Recursion Theory

Set Theory

Computer Science



1. Beth’s Theorem in Cardinality Logics, Israel J. Math., Vol. 14, No. 2, (1973), pp. 205-212.

2. Countable Models of Set Theories, Lecture Notes in Mathematics, Vol. 337, Springer-Verlag, (1973), pp. 539-573.

3. On Existence Proofs of Hanf Numbers, J. of Symbolic Logic, Vol. 39, No. 2, (1974), pp. 318-324.

4. Adding Propositional Connectives to Countable Infinitary Logic, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 77, No. 1, (1975), pp. 1-6.

5. On Decidability of Equational Theories, J. of Pure and Applied Algebra, Vol. 7, (1976), pp. 1-3.

6. The Complexity of Explicit Definitions, Advances in Mathematics, Vol. 20, No. 1, (1976), pp. 18-29.

7. On the Naturalness of Definable Operations, Houston J. Math., Vol. 5, No. 3, (1979), pp. 325-330.

8. (with L. Stanley), A Borel Reducibility Theory for Classes of Countable Structures, J. of Symbolic Logic, Vol. 54, No. 3, September 1989, pp. 894-914.

9.(with Akos Seress), Decidability in Elementary Analysis I, Advances in Math., Vol. 76, No. 1, July 1989, pp. 94-115.

10. (with Akos Seress), Decidability in Elementary Analysis II, Advances in Math., Vol. 79, No. 1, January 1990, pp. 1-17.

11. (with Chris Miller), Expansions of 0-minimal structures by sparse sets, Fundamenta Mathematicae 167 (2001), 55-64.

12. (with Chris Miller), Expansions of O-minimal structures by fast sequences, Journal of Symbolic Logic, 70, June, 2005, pp. 410-418.

13. What is o-minimality?, Logic Colloquium ’06, Special issue of Annals of Pure and Applied Logic, Volume 156, Issue 1, November, 2008, pages 59-67. Logic Colloquium 2006.

14. (with Krzysztof Kurdyka, Chris Miller, and Patrick Speissegger), Expansions of the real field by open sets: definability versus interpretability, Journal of Symbolic Logic, Volume 75, Issue 4 (2010), 1311-1325.


15. Bar Induction and Pi-1-1-CA, J. of Symbolic Logic, Vol. 34, No. 3, (1969), pp. 353-362.

16. Iterated Inductive Definitions and Sigma-1-2-AC, Intuitionism and Proof Theory, North-Holland, (1970), pp. 435-442.

17. The Consistency of Classical Set Theory Relative to a Set Theory with Intuitionistic Logic, J. of Symbolic Logic, Vol. 38, No. 2, (1973), pp. 315-319.

18. Some Applications of Kleene’s Methods for Intuitionistic Systems, Lecture Notes in Mathematics, Vol. 337, Springer-Verlag, (1973), pp. 113-170.

19. The Disjunction Property Implies the Numerical Existence Property, Proc. Natl. Acad. Sci., Communicated by K. Gödel, Vol. 72, No. 8, (August 1975), pp. 2877-2878.

20. Some Systems of Second Order Arithmetic and Their Use, Proceedings of the 1974 International Congress of Mathematicians, Vol. 1, (1975), pp. 235-242.

21. Subsystems of Second Order Arithmetic with Restricted Induction I, II, abstracts, J. of Symbolic Logic, Vol. 41, No. 2, (1976), pp. 557-559.

22. Set Theoretic Foundations for Constructive Analysis, Annals of Mathematics, Vol. 105, (1977), pp. 1-28.

23. On the Derivability of Instantiation Properties, J. of Symbolic Logic, Vol. 42, No. 4, (1977), pp. 506-514.

24. Classically and Intuitionistically Provably Recursive Functions, Higher Set Theory, Springer Lecture Notes, Vol. 669, (1978), pp. 21-27.

25. A Strong Conservative Extension of Peano Arithmetic, Proceedings of the 1978 Kleene Symposium, North Holland, (1980), pp. 113-122.

26. (with K. McAloon and S. Simpson), A Finite Combinatorial Principle Equivalent to the 1-consistency of Predicative Analysis, Patras Logic Symposion, ed. G. Metakides, North-Holland, (1982), pp. 197-230.

27. (with S. Simpson and R. L. Smith), Countable Algebra and Set Existence Axioms, Annals of Pure and Applied Logic, 25 (1983), pp. 141-181.

28. (with A. Scedrov), Set Existence Property for Intuitionistic Theories with Dependent Choice, Annals of Pure and Applied Logic, 25, (1983), pp. 129-140, and corrigendum, 26, (1984), p. 101.

29. (with A. Scedrov), Large Sets in Intuitionistic Set Theory, Annals of Pure and Applied Logic 27 (1984), pp. 1-24.

30. (with A. Scedrov), Arithmetic Transfinite Induction and Recursive Well Orderings, Advances in Math., Vol. 56, No. 3, June 1985, pp. 283-294.

31. (with A. Scedrov), Intuitionistically Provable Recursive Well Orderings, Annals of Pure and Applied Logic, Vol. 30, 1986, pp. 165-171.

32. (with A. Scedrov), The Lack of Definable Witnesses and Provably Recursive Functions in Intuitionistic Set Theories, Advances in Math., Vol. 57, No. 1, July 1985, pp. 1-13.

33. (with A. Scedrov), On the Quantificational Logic of Intuitionistic Set Theory, Math. Proc. of the Cambridge Philosophical Society, Vol. 99, No. 5, 1986, pp. 5-10.

34. (with R. Flagg) Epistemic and Intuitionistic Formal Systems, Annals of Pure and Applied Logic, Vol. 32, 1986, pp. 53-60.

35. (with M. Sheard) An Axiomatic Approach to Self-referential Truth, Annals of Pure and Applied Logic, vol. 33, 1987, 1-21.

36. (with P. Freyd and A. Scedrov), Lindenbaum Algebras of Intuitionistic Theories and Free Categories, Annals of Pure and Applied Logic, vol. 35, 1987, 167-172.

37. (with R. Flagg), Maximality in Modal Logic, Annals of Pure and Applied Logic, vol. 34, 1987, 99-118.

38. (with N. Robertson and P. Seymour), The Metamathematics of the Graph Minor Theorem, Logic and Combinatorics, ed. S. Simpson, AMS Contemporary Mathematics Series, vol. 65, 1987, 229-261.

39. (with M. Sheard), The Equivalence of the Disjunction and Existence Properties for Modal Arithmetic, Journal of Symbolic Logic, Vol. 54, No. 4, December 1989, pp. 1456-1459.

40. (with M. Sheard), The Disjunction and Existence Properties for Axiomatic Systems of Truth, Annals of Pure and Applied Logic 40 (1988), pp. 1-10.

41. (with J. Hirst), Weak comparability of well orderings and reverse mathematics, Annals of Pure and Applied Logic 47 (1990), pp. 11-29.

42. (with J. Hirst), Reverse Mathematics of Homeomorphic Embeddings, Annals of Pure and Applied Logic 54 (1991), pp. 229-253.

43. (with R. K. Meyer), Wither Relevance Arithmetic?, Journal of Symbolic Logic, Vol. 57, No. 3, September 1992, pp. 824-831.

44. (with S. Simpson and X. Yu), Periodic points and subsystems of second order arithmetic, Annals Of Pure and Applied Logic 62 (1993), pp. 51-64.

45. (with M. Sheard), Elementary descent recursion and proof theory, Annals of Pure and Applied Logic 71 (1995), pp. 1-45.

46. (with S. Simpson), Issues and problems in reverse mathematics, in: Computability Theory and Its Applications, Contemporary Mathematics, volume 257, 2000, 127-144.

47. Internal finite tree embeddings, in: Reflections on the Foundations of Mathematics: Essays in honor of Solomon Feferman, ed. Wilfried Sieg, Richard Sommer, Carolyn Talcott, Lecture Notes in Logic, Association for Symbolic Logic, pp. 62-93, AK Peters, 2002.

48. Metamathematics of comparability, in: Reverse Mathematics, ed. S.G. Simpson, Lecture Notes in Logic, vol. 21, ASL, 201-218, 2005.

49. Maximal Nonfinitely Generated Subalgebras, in: Reverse Mathematics, ed. S.G. Simpson, Lecture Notes in Logic, vol. 21, ASL, 189-200, 2005.

50. The Inevitability of Logical Strength: strict reverse mathematics, Logic Colloquium ’06, ASL, ed. Cooper, Geuvers, Pillay, Vaananen, 2009, 373 pages, Cambrdige University Press, pp. 135-183.

51. My Forty Years on His Shoulders. Horizons of Truth, Proceedings of the Goedel Centenary, Cambridge University Press, 339-432, 2011.

52. Remarks on Gödel Phenomena and the Field of Reals, in: Seventy Years of Foundational Studies: A Tribute to Andrzej Mostowski, ed. Ehrenfeucht, Marek, Srebny, IOS Press, 2008, 68-71.

53. Harvey Friedman and Florian Pelupessy, Independence of Ramsey Theorem Variants Using epsilon 0, Proceedings of the AMS, in press, 2015.


54. (with R. Jensen), Note on Admissible Ordinals, The Syntax and Semantics of Infinitary Languages, Springer-Verlag Lecture Notes in Mathematics, Vol. 72, (1968), pp. 77-79.

55. Axiomatic Recursive Function Theory, Logic Colloquium ’69, North-Holland, (1971), pp. 113-137.

56. Algorithmic Procedures, Generalized Turing Algorithms, and Elementary Recursion Theory, Logic Colloquium ’69, North-Holland, (1971), pp. 361-389.

57. (with H. Enderton), Approximating the Standard Model of Analysis, Fundamenta Mathematicae, LXXII, (1971), pp. 175-188.

58. Borel Sets and Hyperdegrees, J. of Symbolic Logic, Vol. 38, No. 3, (1973), pp. 405-409.

59. Minimality in the Delta-1-2-degrees, Fundamenta Mathematicae, LXXXI, (1974), pp. 183-192.

60. Equality Between Functionals, Logic Colloquium, Springer Lecture Notes, Vol. 453, (1975), pp. 22-37.

61. Recursiveness in Pi-1-1 Paths Through O, Proceedings of the AMS, Vol. 54, (January 1976), pp. 311-315.

62. Provable Equality in Primitive Recursive Arithmetic with and without Induction, Pacific J. of Math., Vol. 57, No. 2, (1975), pp. 379-392.

63. Uniformly Defined Descending Sequences of Degrees, J. of Symbolic Logic. Vol. 41, No. 2, (1976), pp. 363-367.

64. (with R. Mansfield), Algorithmic procedures, Transactions of the AMS, vol. 332 (1992), no. 1, pp. 297-312.

65. (with T. Erdelyi), The Number of Certain Integral Polynomials and Nonrecursive Sets of Integers, Part I, Transactions of the AMS, 357, (2005), 999-1011.

66. The Number of Certain Integral Polynomials and Nonrecursive Sets of Integers, Part II, Transactions of the AMS, 357 (2005), 1013-1023.


67. A More Explicit Set Theory, Axiomatic Set Theory, AMS Symposium Pure Mathematics, Vol. XIII, Part I, (1971), pp. 49-65.

68. Higher Set Theory and Mathematical Practice, Annals of Math. Logic, Vol. 2, No. 3, (1971), pp. 325-357.

69. Determinateness in the Low Projective Hierarchy, Fundamenta Mathematicae, LXXII, (1971), pp. 79-95.

70. On Closed Sets of Ordinals, Proceedings of the AMS, (1974), pp. 190-192.

71. PCA Well-orderings of the Line, J. of Symbolic Logic, Vol. 39, No. 1, (1974), pp. 79-80.

72. Large Models of Countable Height, Transactions of the AMS , Vol. 201, (1975), pp. 227-239.

73. A Definable Non-separable Invariant Extension of Lebesgue Measure, Illinois J. Math., Vol. 21, No. 1, (1977), pp. 140-147.

74. Categoricity with Respect to Ordinals, Higher Set Theory, Springer Lecture Notes, Vol. 669, (1978), pp. 17-20.

75. A Proof of Foundation from the Axioms of Cumulation, Higher Set Theory, Springer Lecture Notes, Vol. 669, (1978), pp. 15-16.

76. A Consistent Fubini-Tonelli Theorem for Nonmeasureable Functions, Illinois J. Math., Vol. 24, No. 3, (1980), pp. 390-395.

77. On Definability of Nonmeasurable Sets, Canadian J. Math., Vol. XXXII, No. 3, (1980), pp. 653-656.

78. (with M. Talagrand), Un Ensemble Singulier, Bull. Sci. Math. 104, (1980), pp. 337-340.

79. On the Necessary Use of Abstract Set Theory, Advances in Math., Vol. 41, No. 3, September 1981, pp. 209-280.

80. Unary Borel Functions and Second Order Arithmetic, Advances in Math., Vol. 50, No. 2, November 1983, pp. 155-159.

81. Necessary Uses of Abstract Theory in Finite Mathematics, Advances in Math., Vol. 60, No. 1, 1986, 92-122.

82. Finite Functions and the Necessary Use of Large Cardinals, Annals of Math., Vol. 148, No. 3, 1998, pp. 803–893.

83. Borel and Baire reducibility, Fundamenta Mathematicae, 164 (2000), 61-69.

84. Subtle cardinals and linear orderings, Annals of Pure and Applied Logic 107 (2001), 1-34.

85. Selection for Borel relations, in: Logic Colloquium 01, ed. J Krajicek, Lecture Notes in Logic, volume 20, ASL, 2005, 151-169.

86. Primitive Independence Results, Journal of Mathematical Logic, Volume 3, Number 1, May 2003, 67-83.

87. Three quantifier sentences, Fundamenta Mathematicae, 177 (2003), 213-240.

88. A Way Out, in: One Hundred Years of Russell’s Paradox, ed. Godeharad Link, de Gruyter, 49-86, 2004.

89. Working with Nonstandard Models, in: Nonstandard Models of Arithmetic and Set Theory, American Mathematical Society, ed. Enayat and Kossak, 71-86, 2004.

90. Invariant Maximality and Incompleteness, in: Foundations and Methods from Mathematics to Neuroscience: Essays Inspired by Patrick Suppes. (2014) Edited by Colleen E. Crangle, Adolfo Garcia de la Sienra, and Helen E. Longino. Stanford, California: CSLI Publications.

91.  Concrete Mathematical Incompleteness: Basic Emulation Theory, Chapter 12 in Hilary Putnam on Logic and Mathematics, ed. Geoffrey Hellman and Roy T. Cook, Outstanding Contributions to Logic; 9. Springer, 2018. ISBN 978-3-319-96273-3 (hbk); 978-3-319-96274-0 (e-book). Pp. x + 274.


92. (with Ker-I Ko), Computational Complexity of Real Functions, J. of Theoretical Comp. Science, 20, (1982), pp. 323-352.

93. The Computational Complexity of Maximization and Integration, Advances in Math., Vol. 53, No. 1, 1984, pp. 80-98.

94. On the Spectra of Universal Relational Sentences, Information and Control, Vol. 62, No. 23, August/September 1984, pp. 204-209.

95. (with Ker-I Ko), Computing Power Series in Polynomial Time, Advances in Applied Mathematics, 9, 40-50 (1988).

96. Applications of Mathematics to Computer Science, in: Emerging syntheses in science, ed. by David Pines, Proceedings of the Founding Workshops of the Santa Fe Institute, volume 1, 1988, pp. 205 – 210.

97. (with R. Flagg), A Framework for Measuring the Complexity of Mathematical Concepts, Advances in Applied Mathematics, volume 11, no. 1, March, 1990, pp. 1-34.

98. Some Decision Problems of Enormous Complexity, Proceedings of the 14th Annual IEEE Symposium on Logic in Computer Science, July 2-5, 1999, Trento, Italy, pp. 2-12.

99. (with J. Avigad), Combining decision procedures for the reals, Logical Methods in Computer Science 2(4:4), 2006.

100. (with S. Kieffer, J. Avigad), A language for mathematical knowledge management, in: Computer Reconstruction of the Body of Mathematics, ed. A. Grabowski, A. Naumowicz, Studies in Logic, Grammar and Rhetoric, p. 51-66, 2009.

101. Sitaraman, M., Adcock, B., Avigad, J., Bronish, D., Bucci, B., Frazier, D., Friedman, H.M., Harton, Heym, W., Kirschenbaum, J., Krone, J., Smith, H., and Weide, B.W., “Building a Push-Button RESOLVE Verifier: Progress and Challenges”, Formal Aspects of Computing 23, 5 (2011), 607-626.

102. Zaccai, D., Tagore, A., Hoffman, D., Kirschenbaum, J., Bainazarov, Z., Friedman, H.M., Pearl, D.K., and Weide, B.W., “Syrus: Providing Practice Problems in Discrete Mathematics With Instant Feedback”, Proceedings of the 45th SIGCSE Technical Symposium on Computer Science Education, ACM Press, March 2014, to appear.


103. One Hundred and Two Problems in Mathematical Logic, J. of Symbolic Logic, Vol. 40, No. 2, (1975), pp. 113-129.

104. A Cumulative Hierarchy of Predicates, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, Bd. 21, (1975), pp. 309-314.

105. The Incompleteness Phenomena, Proceedings of the AMS Centennial Celebration, AMS Centennial Publications, Volume II, Mathematics into the Twenty-first Century, 1992, pp. 49-84.

106. Long finite sequences, Journal of Combinatorial Theory, Series A 95, 102-144 (2001).

107. (with Solomon Feferman, Penelope Maddy, and John Steel), Does mathematics need new axioms?, The Bulletin of Symbolic Logic, Volume 6, Number 4, December 2000, 401-446.

108. Concept Calculus: Much Better Than, in: New Frontiers in Research on Infinity, ed. Michael Heller and W. Hugh Woodin, Cambridge University Press, 130-164, 2011.

109. Limitations on our Understanding of the Behavior of Simplified Physical Systems, Proceedings of the 2007 San Marino Symposium on “Science Reason and Truth”; Ed. by M. Bersanelli, C. Harper and P. Van Inwagen, Euresis Journal, Volume 5, Summer 2013, p. 125- 153.

110. (with Ovidiu Costin), Foundational aspects of singular integrals, Journal of Functional Analysis 267 (2014) 4732–4752.


A Divine Consistency Proof for Mathematics, Notre Dame Journal of Formal Logic, December 25, 2012.

(with Florian Pelupessy), Independence of Ramsey Theorem Variants Using Î0, February 7, 2013.

(with Albert Visser), When Bi-Interpretability Implies Synonymy, January 3, 2014.

Concept Calculus: universes, October 2, 2012. Available at

Unique Undefinable Elements, January 9, 2013. Available at


Boolean Relation Theory and Incompleteness, Lecture Notes in Logic, Association for Symbolic Logic, to appear, 2021.  Available at