Model Theory

Proof Theory and Intuitionism

Recursion Theory

Set Theory

Computer Science

Miscellaneous

**MODEL THEORY**

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.

*PROOF THEORY AND INTUITIONISM*

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. Independence of Ramsey Theorem Variants Using epsilon 0, Proceedings of the AMS, in press, 2015.

**RECURSION THEORY**

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.

**SET THEORY**

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.

**COMPUTER SCIENCE**

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.

**MISCELLANEOUS**

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.

**PAPERS SUBMITTED**

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

(with Florian Pelupessy), Independence of Ramsey Theorem Variants Using Î_{0}, February 7, 2013.

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

Concept Calculus: universes, October 2, 2012. Available at http://www.math.ohio-state.edu/%7Efriedman/index.html

Unique Undefinable Elements, January 9, 2013. Available at http://www.math.ohio-state.edu/%7Efriedman/index.html

**BOOK TO APPEAR**

Boolean Relation Theory and Incompleteness, Lecture Notes in Logic, Association for Symbolic Logic, to appear, 2021. Available at http://www.math.ohio-state.edu/%7Efriedman/index.html