"The Current Status of the Foundations of Mathematics"

Compiled by

Eddie Yeghiayan

(with Ronald Jensen.) "Notes on Admissible Ordinals." In Jon Barwise, ed.,

*The Syntax and Semantics of Infinitary Languages*, pp. 77-79. Lecture Notes in Mathematics, 72. Berlin & New York: Springer, 1968."Subsystems of Analysis." PhD Dissertation, MIT, 1968.

"Bar Induction and Pi-1-1-CA."

*Journal of Symbolic Logic*(September 1969), 34(3):353-362.

"Iterated Inductive Definitions and Sigma 1-2-AC." In Akino Kino, John Myhill and Richard Vesley, eds.,

*Intuitionism and Proof Theory. Proceedings of the Summer Conference at Buffalo, N.Y. 1968*, pp. 435-442. Studies in Logic and the Foundations of Mathematics. Amsterdam: North-Holland, 1970.

Proceedings of the Conference on Intuitionism and Proof Theory held at the State University of New York at Buffalo, in 1968.

"Algorithmic Procedures, Generalized Turing Algorithms, and Elementary Recursion Theory." In R.O. Gandy and C.M.E. Yates, eds.,

*Logic Colloquium '69. Proceedings of the Summer School and Colloquium in Mathematical Logic, Manchester, August 1969*, pp. 361-389. Studies in Logic and the Foundations of Mathematics, 61. Amsterdam: North-Holland, 1971.(with H.B. Enderton.) "Approximating the Standard Model of Analysis."

*Fundamenta Mathematicae*(1971), 72(2):175-188."Axiomatic Recursive Function Theory." In R.O. Gandy and C.M.E. Yates, eds.,

*Logic Colloquium '69. Proceedings of the Summer School and Colloquium in Mathematical Logic, Manchester, August 1969*, pp. 113-137. Studies in Logic and the Foundations of Mathematics, 61. Amsterdam: North-Holland, 1971."Determinateness in the Low Projective Hierarchy."

*Fundamenta Mathematicae*(1971), 72(1):79-95."Higher Set Theory and Mathematical Practice."

*Annals of Mathematical Logic*(January 1971), 2(3):325-257."A More Explicit Set Theory." In Dana S. Scott, ed.,

*Axiomatic Set Theory*, pp. 49-65. Proceedings of Symposia in Pure Mathematics, 13, Part I. Providence, RI: American Mathematical Society, 1971.

Held at the University of California, Los Angeles, California, July 10-August 5, 1967.

"Beth's Theorem in Cardinality Logics."

*Israel Journal of Mathematics*(1973), 14(2):205-212."Borel Sets and Hyperdegrees."

*Journal of Symbolic Logic*(December 1973), 38(3):405-409."The Consistency of Classical Set Theory Relative to a Set Theory with Intuitionistic Logic."

*Journal of Symbolic Logic*(June 1973), 38(2):315-319."Countable Models of Set Theories." In A. R. D. Mathias and Hartley Rogers, eds.,

*Cambridge Summer School in Mathematical Logic: Papers*, pp. 539-573. Lecture Notes in Mathematics, 337. Berlin & New York, Springer, 1973.

Held in Cambridge, England, August 1-21, 1971."Some Applications of Kleene's Methods for Intuitionistic Systems." In A. R. D. Mathias and Hartley Rogers, eds.,

*Cambridge Summer School in Mathematical Logic: Papers*, pp. 113-170. Lecture Notes in Mathematics, 337. Berlin & New York: Springer, 1973.

"Minimality in the Delta-1-2-degrees."

*Fundamenta Mathematicae*(1974), 81(3):183-192.

Collection of articles dedicated to Andrzej Mostowski on the occasion of his sixtieth birthday, III."On Closed Sets of Ordinals."

*Proceedings of the American Mathematical Society*(March 1974), 43(1):190-192."On Existence Proofs of Hanf Numbers."

*Journal of Symbolic Logic*(June 1974), 39(2):318-324."PCA Well-Orderings of the Line."

*Journal of Symbolic Logic*(March 1974), 39(1):79-80.

"Adding Propositional Connectives to Countable Infinitary Logic."

*Mathematical Proceedings of the Cambridge Philosophical Society*(January 1975), 77(1):1-6."A Cumulative Hierarchy of Predicates."

*Zeitschrift für mathematische Logik und Grundlagen der Mathematik*(1975), 21(4):309-314."The Disjunction Property Implies the Numerical Existence Property."

*Proceedings of the National Academy of Sciences of the United States of America*(August 1975), 72(8):2877-2878."Equality Between Functionals." In Rohit Parikh, ed.,

*Logic Colloquium: Symposium on Logic held at Boston, 1972-73*, pp. 22-37. Lecture Notes in Mathematics, 453. Berlin & New York: Springer, 1975."Large Models of Countable Height."

*Transactions of the American Mathematical Society*(January 1975), 201(474):227-239."One Hundred and Two Problems in Mathematical Logic."

*Journal of Symbolic Logic*(June 1975), 40(2):113-129."Provable Equality in Primitive Recursive Arithmetic With and Without Induction."

*Pacific Journal of Mathematics*(April 1975), 57(2):379-392."Some Systems of Second Order Arithmetic and Their Use." In Ralph D. James, ed.,

*Proceedings of the International Congress of Mathematicians*, Vol. 1, pp. 235-242. Vancouver: Canadian Mathematical Congress, 1975.

Proceedings of the International Congress of Mathematicians held in 1974 in Vancouver, British Columbia.

"The Complexity of Explicit Definitions."

*Advances in Mathematics*(April 1976), 20(1):18-29."On Decidability of Equational Theories."

*Journal of Pure and Applied Algebra*(January 1976), 7(1):1-3."Recursiveness in Pi-1-1 Paths Through O."

*Proceedings of the American Mathematical Society*(January 1976), 54(1) [199]:311-315."Subsystems of Second Order Arithmetic with Restricted Induction. I." [Abstract]

*Journal of Symbolic Logic*(June 1976), 41(2):557-558."Subsystems of Second Order Arithmetic with Restricted Induction. II." [Abstract]

*Journal of Symbolic Logic*(June 1976), 41(2):558-559."Uniformly Defined Descending Sequences of Degrees."

*Journal of Symbolic Logic*(June 1976), 41(2):363-367.

"A Definable Nonseparable Invariant Extension of Lebesgue Measure."

*Illinois Journal of Mathematics*(March 1977), 21(1) [82]:140-147."On the Derivability of Instantiation Properties."

*Journal of Symbolic Logic*(December 1977), 42(4):506-514.

**Abstract**:"Every recursively enumerable extension of arithmetic which obeys the disjunction property obeys the numerical existence property $$LEFT BRACKET$$FR, 1 $$RIGHT BRACKET$$. The requirement of recursive enumerability is essential. for extensions of intuitionistic second order arithmetic by means of sentences (in its language) with no existential set quantifiers, the numerical existence property implies the set existence property. the restriction on existential set quantifiers is essential. The numerical existence property cannot be eliminated, but in the case of finite extensions of has, can be replaced by a weaker form of it. As a consequence, the set existence property for intuitionistic second order arithmetic can be proved within itself.""Set Theoretic Foundations for Constructive Analysis."

*Annals of Mathematics*[2nd Series] (January 1977), 105(1):1-28.

"Categoricity with Respect to Ordinals." In G. H. Muller and D. S. Scott, eds.,

*Higher Set Theory: Proceedings, Oberwolfach, Germany, April, 13-23, 1977*, pp. 17-20. Lecture Notes in Mathematics, 669. New York & Berlin: Springer, 1978."Classically and Intuitionistically Provably Recursive Functions." In G. H. Muller and D. S. Scott, eds.,

*Higher Set Theory: Proceedings, Oberwolfach, Germany, April, 13-23, 1977*, pp. 21-27. Lecture Notes in Mathematics, 669. Berlin & New York: Springer, 1978."A Proof of Foundation from Axioms of Cumulation." In G. H. Muller and D. S. Scott, eds.,

*Higher Set Theory: Proceedings, Oberwolfach, Germany, April, 13-23, 1977*, pp. 15-16. Lecture Notes in Mathematics, 669. Berlin & New York: Springer, 1978.

"On the Naturalness of Definable Operations."

*Houston Journal of Mathematics*(1979), 5(3):325-330.

"A Consistent Fubini-Tonelli Theorem for Nonmeasurable Functions."

*Illinois Journal of Mathematics*(Fall 1980), 24(3) [96]:390-395.(with Michel Talagrand.) "Un ensemble singulier."

*Bulletin des Sciences Mathématiques*[2nd Series] (1980), 104(4):337-340."On Definability of Nonmeasurable Sets."

*Canadian Journal of Mathematics*(June 1980), 32(3):653-656."A Strong Conservative Extension of Peano Arithmetic." In Jon Barwise, H. Jerome Keisler, and Kenneth Kunen eds.,

*The Kleene Symposium: Proceedings of the Symposium held June 18-24, 1978 at Madison, Wisconsin, U.S.A.*, pp. 113-122. Studies in Logic and the Foundations of Mathematics, 101. Amsterdam & New York: North-Holland, 1980.

**Abstract**: "A system of Alpo of set theory is presented and proved to form a conservative extension of Peano arithmetic. The system has sufficient strength to allow a very substantial amount of analysis to be directly formalized, including Lebesgue integration theory, without resorting to wholesale padding of objects with extra information."

- "On the Necessary Use of Abstract Set Theory."
*Advances in Mathematics*(September 1981), 41(3):209-280.

(with Ker I. Ko.) "Computational Complexity of Real Functions."

*Theoretical Computer Science*(1982), 20(3):323-352.(with Kenneth McAloon and Stephen G. Simpson.) "A Finite Combinatorial Principle Equivalent to the 1-Consistency of Predicative Analysis." In George Metakides, ed.,

*Patras Logic Symposion: Proceedings of the Logic Symposion held at Patras, Greece, August 18-22, 1980*, pp. 197-230. Studies in Logic and the Foundations of Mathematics, 109. Amsterdam & New York: North-Holland, 1982.

(with Stephen G. Simpson and Rick L. Smith.) "Countable Algebra and Set Existence Axioms."

*Annals of Pure and Applied Logic*(1983), 25(2):141-181.(with Andrej Scedrov.) "Set Existence Property for Intuitionistic Theories with Dependent Choice."

*Annals of Pure and Applied Logic*(1983), 25(2):129-140.

**Abstract**: "Let TC be intuitionistic higher-order arithmetic or intuitionistic ZF(with replacement), both with relativized dependent choice, or just countable choice. We show that TC(T(EX. A("X") (Closed) Gives TC(TA(T) for some comprehension term T. This solves a problem left open by Myhill in (4).""Unary Borel Functions and Second-Order Arithmetic."

*Advances in Mathematics*(October 1983), 50(2):155-159.

"The Computational Complexity of Maximation and Integration."

*Advances in Mathematics*(July 1984), 53(1):80-98."Correction."

*Annals of Pure and Applied Logic*(1984), 26(1):101.

For "Set Existence Property for Intuitionistic Theories with Dependent Choice" (1983).(with Andrej Scedrov.) "Large Sets in Intuitionistic Set Theory."

*Annals of Pure and Applied Logic*(1984), 27(1):1-24."On the Spectra of Universal Relational Sentences."

*Information and Control*(August-September 1984), 62(2-3):205-209.

(with Stephen G. Simpson and Rick L. Smith.) "Addendum to 'Countable Algebra and Set Existence Axioms'."

*Annals of Pure and Applied Logic*(1985), 28(3):319-320.

See "Countable Algebra and Set Existence Axioms" (1983).(with Andrej Scedrov.) "Arithmetic Transfinite Induction and Recursive Well-Orderings."

*Advances in Mathematics*(June 1985), 56(3):283-294.(with Andrej Scedrov.) "The Lack of Definable Witnesses and Provably Recursive Functions in Intuitionistic Set Theories."

*Advances in Mathematics*(July 1985), 57(1):1-13.

(with R.C. Flagg.) "Epistemic and Intuitionistic Formal Systems."

*Annals of Pure and Applied Logic*(September 1986), 32(1):53-60.(with Andre Scedrov.) "Intuitionistically Provable Recursive Well-Orderings."

*Annals of Pure and Applied Logic*(1986), 30(2):165-171."Necessary Uses of Abstract Set Theory in Finite Mathematics."

*Advances in Mathematics*(April 1986), 60(1):92-122.(with Andrej Scedrov.) "On the Quantificational Logic of Intuitionistic Set-Theory."

*Mathematical Proceedings Cambridge Philosophical Society*(January 1986), 99(1):5-10.

(with Michael Sheard.) "An Axiomatic Approach to Self-Referential Truth."

*Annals of Pure and Applied Logic*(January 1987), 33(1):1-21.

**Abstract**: "We add a new predicate T to the language of Peano Arithmetic, with T(x) intended to mean '"x" is the Gödel number of a true sentence of the augmented language'. We create a list of plausible axioms and rules of inference concerning this predicate T, each of which embodies some aspect of its intended interpretation as truth. We classify all subsets of the list as either consistent or inconsistent, and we measure the proof-theoretic strength of several subsets by comparing them with familiar system of arithmetic and analysis."(with Peter Freyd and Andre Scedrov.) "Lindenbaum Algebras of Intuitionistic Theories and Free Categories."

*Annals of Pure and Applied Logic*(August 1987), 35(2):167-172.(with R.C. Flagg.) "Maximality in Modal Logic."

*Annals of Pure and Applied Logic*(May 1987), 34(2):99-118.(with Neil Robertson and Paul Seymour.) "The Metamathematics of the Graph Minor Theorem." In Stephen G. Simpson, ed.,

*Logic and Combinatorics: Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference held August 4-10, 1985*, pp. 229-261. Contemporary Mathematics, 65. Providence, RI: American Mathematical Society, 1987.

Held at Humboldt State University in Arcata, Calif.

"Applications of Mathematics to Computer Science." In David Pines, ed.,

*Emerging Syntheses in Science: Proceedings of the Founding Workshops of the Santa Fe Institute, Santa Fe, New Mexico*, pp. 205-210. Proceedings Volume in the Sante Fe Institute Studies in the Sciences of Complexity, 1. Redwood City, Calif.: Addison-Wesley, 1988.(with Ker-I. Ko.) "Computing Power Series in Polynomial Time."

*Advances in Applied Mathematics*(1988), 9(1):40-50.(with Michael Sheard.) "The Disjunction and Existence Properties for Axiomatic Systems of Truth."

*Annals of Pure and Applied Logic*(October 1988), 40(1):1-10.

(with Stanley Lee.)"A Borel Reducibility Theory for Classes of Countable Structures."

*Journal of Symbolic Logic*(September 1989), 54(3):894-914.

**Abstract**: "We introduce a reducibility preordering between classes of countable structures, each class containing only structures of a given similarity type (which is allowed to vary from class to class). though we sometimes work in a slightly larger context, we are principally concerned with the case where each class is an invariant borel class (i.e., the class of all models, with underlying set = "W", of an "L"("W" $$SUBSCRIPT$$1 ,"W") sentence; from this point of view, the reducibility can be thought of as a (rather weak) sort of "L"("W" $$SUBSCRIPT$$1 ,"W")-Interpretability notion). We prove a number of general results about this notion, but our main thrust is to situate various mathematically natural classes with respect to the preordering, most notably classes of algebraic structures such as groups and fields."(with Ákos Seress.) "Decidability in Elementary Analysis. I."

*Advances in Mathematics*(July 1989), 76(1):94-115.(with Michael Sheard.) "The Equivalence of the Disjunction and Existence Properties for Modal Arithmetic."

*Journal of Symbolic Logic*(December 1989), 54(4):1456-1459.

(with Ákos Seress.) "Decidability in Elementary Analysis. II."

*Advances in Mathematics*(January 1990), 79(1):1-17.(with R. C. Flagg.) "A Framework for Measuring the Complexity of Mathematical Concepts."

*Advances in Applied Mathematics*(March 1990), 11(1):1-34.(with Jeffry L. Hirst.) "Weak Comparability of Well Orderings and Reverse Mathematics."

*Annals of Pure and Applied Logic*(April 18, 1990), 47(1):11-29.

(with Jeffry L. Hirst.) "Reverse Mathematics and Homeomorphic Embeddings."

*Annals of Pure and Applied Logic*(November 11, 1991), 54(3):229-253.

(with Richard Masefield.) "Algorithmic Procedures."

*Transactions of the American Mathematical Society*(July 1992), 332(1):297-312.

Text from*JSTOR*"The Incompleteness Phenomena." In Felix E. Browder, ed.,

*American Mathematical Society Centennial Publications*, Vol. II: Mathematics into the Twenty-first Century, pp. 49-84. Providence, RI: American Mathematical Society, 1992.(with Robert K. Meyer.) "Whither Relevant Arithmetic?"

*Journal of Symbolic Logic*(September 1992), 57(3):824-831.

(with Stephen G. Simpson and Xiaokang Yu.) "Periodic Points and Subsystems of Second-Order Arithmetic."

*Annals of Pure and Applied Logic*(June 28, 1993), 62(1):51-64.

(with Michael Sheard.) "Elementary Descent Recursion and Proof Theory."

*Annals of Pure and Applied Logic*(January 15, 1995), 71(1):1-45.

**Abstract**: "We define a class of functions, the*descent recursive*functions, relative to an arbitrary elementary recursive system of ordinal notations. By means of these functions, we provide a general technique for measuring the proof-theoretic strength of a variety of systems of first- order arithmetic. We characterize the provable well- orderings and provably recursive functions of these systems, and derive various conservation and equiconsistency results.""Some Decision Problems of Enormous Complexity." In IEEE Computer Society Technical Committee on Mathematical Foundations of Computing,

*14th Symposium on Logic in Computer Science Proceedings: July 2-5, 1999, Trento, Italy*, pp. 2-12. Los Alamitos, Calif.: IEEE Computer Society Press, 1999.

Online: http://ieeexplore.ieee.org/lpdocs/epic03/RecentCon.htm?punumber=6352

"Finite Functions and the Necessary Use of Large Cardinals."

*Annals of Mathematics*(November 1998), 148(3):803-893.

- "Borel and Baire Reducibility."
*Fundamenta Mathematicae*(2000), 164(1):61-69. (with Solomon Feferman and Penelope Maddy, and others.) "Does Mathematics Need New Axioms?"

*Bulletin of Symbolic Logic*(December 2000), 6(4):401-446.

Includes Harvey Friedman's "Normal Mathematics Will Need New Axioms," pp. 434-446.(with Stephen G. Simpson.) "Issues and Problems in Reverse Mathematics." In Peter A. Cholak, et al., eds.,

*Computability Theory and its Applications: Current Trends and Open Problems: Proceedings of a 1999 AMS-IMS-SIAM Joint Summer Research Conference, Computability Theory and Applications, June 13-17, 1999, University of Colorado, Boulder*, pp. 127-144. Contemporary Mathematics, 257. Providence, RI: American Mathematical Society, 2000.

(with Chris Miller.) "Expansions of o-Minimal Structures by Sparse Sets."

*Fundamenta Mathematicae*(2001), 167(1):55-64."Long Finite Sequences."

*Journal of Combinatorial Theory*(Series A) (July 2001), 95(1):102-144.

Text"Subtle Cardinals and Linear Orderings."

*Annals of Pure and Applied Logic*(January 15, 2001) 107(1-3):1-34.

**Abstract**: "The subtle, almost ineffable, and ineffable cardinals were introduced in an unpublished 1971 manuscript of R. Jensen and K. Kunen. The concepts were extended to that of k-subtle, k-almost ineffable, and k-ineffable cardinals in 1975 by J. Baumgartner. In this paper we give a self contained treatment of the basic facts about this level of the large cardinal hierarchy, which were established by J. Baumgartner. In particular, we give a proof that the k-subtle, k-almost ineffable, and k-ineffable cardinals define three properly intertwined hierarchies with the same limit, lying strictly above 'total indescribability' and strictly below 'arrowing'. The innovation here is presented in Section 2, where we take a distinctly minimalist approach. Here the subtle cardinal hierarchy is characterized by very elementary properties that do not mention closed unbounded or stationary sets. This development culminates in a characterization of the hierarchy by means of a striking universal second-order property of linear orderings (k-critical)."

"Finite Trees and the Necessay Use of Large Cardinals."

Cubric, Djordje. "On the Semantics of the Universal Quantifier."

*Annals of Pure and Applied Logic*(October 17, 1997), 87(3):209-239.

**Abstract**: "We investigate the universal fragment of intuitionistic logic focussing on equality of proofs. We give categorical models for that and prove several completeness results. One of them is a generalization of the well known Yoneda lemma and the other is an extension of Harvey Friedman's completeness result for typed lambda calculus."Gallier, Jean H. "What's So Special about Kruskal Theorem and the Ordinal Gamma-0 - A Survey of Some Results in Proof Theory."

*Annals of Pure and Applied Logic*(September 19, 1991) 53(3:199-260.Harrington, Leo A., M.D. Morley, Andrej Scedrov, and Stephen G. Simpson, eds.,

*Harvey Friedman's Research on the Foundations of Mathematics*, 137-159, Studies in Logic and the Foundations of Mathematics, 117. Amsterdam & New York: North-Holland, 1985.

"This volume discusses various aspects of Friedman's research in the foundations of mathematics over the past fifteen years. We felt that it was especially worthwhile to present this volume to a wide audience of mathematicians, computer scientists, and mathematically oriented philosophers." (Preface)**Contents**:

Harrington, Leo A., M. D. Morley, Andrej Scedrov, and Stephen G. Simpson. "Introduction":vii-xii.

"Biography of Harvey Friedman":xiii.

Nerode, Anil and Leo A. Harrington. "The Work of Harvey Friedman":1-10.[*Notices American Mathematical Society*(1984), 31(6):563-566.]

Stanley, Lee J. "Borel Diagonalization and Abstract Set theory: Recent Results of Harvey Friedman":11-86.

Simpson, Stephen G. "Nonprovability of Certain Combinatorial Properties of Finite Trees":87-117.

Smith, Rick L. "The Consistency Strengths of Some Finite Forms of the Higman and Kruskal Theorems":119-136.

Simpson, Stephen G. "Friedman's Research on Subsystems of Second-Order Arithmetic":137-159.

Steinhorn, Charles. "Borel Structures for First-Order and Extended Logics":161-178.

Smorynski, C. "Nonstandard Models and Related Developments":179-229.

Leivant, Daniel. "Intuitionistic Formal Systems":231-255.

Scedrov, Andrej. "Intuitionistic Set Theory":257-284.

Shepherdson, J.C. "Algorithmic Procedures, Generalized Turing Algorithms, and Elementary Recursion Theory":285-308.

Shepherdson, J.C. "Computational Complexity of Real Functions":309-315.

Kfoury, A. J. "The Pebble Game and Logics of Programs":317-329.

Statman, R. "Equality between Functionals Revisited":331-338.

Byerly, Robert E. "Mathematical Aspects of Recursive Function Theory":339-352.

Smorynski, C. "'Big' News from Archimedes to Friedman":353-366. [*Notices American Mathematical Society*(1983), 30(3):251-256.]

Smorynski, Craig. "Some Rapidly Growing Functions:367-380. [*Mathematical Intelligencer*(1979-1980),2(3):149-154.]

Smorynski, C. "The Varieties of Arboreal Experience":381-397. [*Mathematical Intelligencer*(1982), 4(4):182-189.]

Kolata, Gina. "Does Gödel's Theorem Matter to Mathematics?":399-404.[*Science*(1982), 218(4574):779-780.]

"Harvey Friedman's Publications:405-408.Heppenheimer, T. A. "The Long Shadow of Kurt Gödel."

*National Science Foundation Mosaic*(Spring 1990), 21(1):2-13.Kolata, Gina. "Does Gödel's Theorem Matter to Mathematics?"

*Science*(November 19, 1982), 218[4574]:779-780.

"The recent discovery of two natural but undecidable statements indicates that Gödel's theorem is more than just a logician's trick."Kris, Igor. "The Structure of Infinite Friedman Trees."

*Advances in Mathematics*(September 15, 1995), 115(1):141-199.Mills, George. "A Tree Analysis of Unprovable Combinatorial Statements." In L. Pacholski, J. Wierzejewski, and A.J. Wilkie, eds.,

*Model Theory of Algebra and Arithmetic: Proceedings of the Conference on Applications of Logic to Algebra and Arithmetic held at Karpacz, Poland, September 1-7, 1979*, pp. 248-311. Springer Lecture Notes, 834. Berlin & New York: Springer, 1980.Nerode, Anil and Leo A. Harrington. "The Work of Harvey Friedman."

*Notices American Mathematical Society*(1984), 31:563-566.Pudlák, Pavel. "Improved Bounds to the Length of Proofs of Finitistic Consistency Statements." In Stephen G. Simpson, ed.,

*Logic and Combinatorics: Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference held August 4-10, 1985*, pp. 309-332. Contemporary Mathematics (American Mathematical Society), 65. Providence, RI: American Mathematical Society, 1987.

Held at Humboldt State University.Ranada, David. "Untangling Musicality: A Computerized Aid to the Analysis of Interpretation."

*Musical America*(January 1991), 111(1):91.Riecke Jon G. "Statmans 1-Section Theorem."

*Information and Computation*(February 1, 1995), 116(2):294-303.Simpson, Steven. "Reverse Mathematics."

*1982 AMS Symp. Pure Math.*Cornell University, to appear.Simpson, Steven. "Sigma-1-1 and Pi-1-1 Transfinite Induction." In D. van Dalen, D. Lascar, T.J. Smiley, eds.,

*Logic Colloquium '80: Papers Intended for the European Summer Meeting of the Association for Symbolic Logic*, pp. 239-253. Amsterdam & New York: North-Holland, 1982.Simpson, Steven. "Which Set Existence Axioms are Needed to Prove the Cauchy/Peano Theorem for Ordinary Differential Equations?"

*Journal of Symbolic Logic*(September 1984), 49(3):783-802.Smorynski, Craig. "'Big' News from Archimedes to Friedman."

*Notices of the American Mathematical Society*(April 1983), 30(3):251-256.Smorynski, Craig. "The Varieties of Arboreal Experience."

*Mathematical Intelligencer*(1982), 4(4):182-189.Steinhorn, Charles. "Borel Structures and Measure and Category Logics." In Jon Barwise and Solomon Feferman, eds.,

*Model-Theoretic Logics*. Chapter XVI. Perspectives in Mathematical Logic. New York: Springer, 1985.

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