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1. Preprints, Drafts, and Abstracts

  1. The Reverse Mathematics program originated with the publications 'Some Systems of Second Order Arithmeitc aand Their Use, Proceedings of the 1974 International Congress of Mathematicians, Vol. 1, (1975), pp. 235-242', and 'Subsystems of Second Order Arithmetic with Restricted Induction I,II, abstracts, J. of Symbolic Logic, Vol. 41, No. 2 (1976), pp. 557-559, as well as the following widely distributed unpublished manuscripts. PDF 'The Analysis Of Mathematical Texts, And Their Calibration In Terms Of Intrinsic Strength I', April 3, 1975, 7 pages. PDF 'The Analysis Of Mathematical Texts, And Their Calibration In Terms Of Intrinsic Strength II', April 8, 1975, 5 pages. PDF 'The Analysis Of Mathematical Texts, And Their Calibration In Terms Of Intrinsic Strength III ', May 19, 1975, 26 pages. PDF 'The Analysis Of Mathematical Texts, And Their Calibration In Terms Of Intrinsic Strength IV', August 15, 1975, 32 pages. PDF 'The Logical Strength Of Mathematical Statements', October 15, 1975, 1 page. PDF 'The Logical Strength Of Mathematical Statements I', August, 1976, 20 pages. See also: (with S. Simpson), Issues and Problems in Reverse Mathematics, in: Computability Theory and its Applications, Contemporary Mathematics, volume 257, 2000, 127-144.
  2. PSPDFDOC file`Combinatorial set theoretic statements of great logical strength', 1995, 4 pages, abstract.
  3. PostScriptDOC file`Some Decision Problems Related to Hilbert's Tenth Problem', January 15, 1996, 2 pages, abstract.
  4. PSPDFDOC file`Extremely large cardinals in the rationals', August 1996, 4 pages, abstract.
  5. PSPDFDOC file`Countable model theory and large cardinals', September 1996, 3 pages, abstract.
  6. PostScriptDOC`On Expansions of o-Minimal Structures', November 23, 1996, 4 pages, draft.
  7. PSPDFDOC file`The Formalization of Mathematics', May 21, 1997, 11 pages, draft.
  8. PSPDFDOC file`Some Historical Perspectives on Certain Incompleteness Phenomena', May 21, 1997, 5 pages, draft.
  9. PSPDFDOC file`Transfer Principles in Set Theory', May 21, 1997, 44 pages, draft.
  10. PDFDOC file'The interpretation of set theory in pure predication theory', August 20, 1997, 64 pages, draft.
  11. PSPDFDOC file`Applications of Large Cardinals to Graph Theory', October 23, 1997, 36 pages, draft.
  12. PSPDFDOC file`The interpretation of set theory in mathematical predication theory', October 25, 1997, 10 pages, draft.
  13. PSPDFDOC file`The axiomatization of set theory by extensionality, separation, and reducibility', October 28, 1997, 71 pages, draft.
  14. PSPDFDOC file`Applications of Large Cardinals to Borel Functions', November 26, 1997, 3 pages, abstract.
  15. PSPDFDOC file`Finite Trees and the Necessary Use of Large Cardinals', March 22, 1998, 58 pages, draft.
  16. PSPDFDOC file`'Subtle Cardinals and Linear Orderings', April 1998, 52 pages, See Publications.
  17. PDFDOC file'Long Finite Sequences', October 8, 1998, 50 pages. See Publications.
  18. PDFDOC file 'Interpretability and Definability in Expansions,' with Chris Miller, November, 1998, 5 pages, abstract.
  19. DOC fileDOC fileDOC file'Some Decision Problems of Enormous Complexity', May 6, 1999, 10 pages, See Publications.
  20. PostScriptDOC`A Consistency Proof for Elementary Algebra and Geometry', August 23, 1999, 38 pages, draft.
  21. `Borel and Baire reducibility', September 2, 1999, 11 pages. See Publications.
  22. PostScriptDOC`Conservation', September 24, 1999, 7 pages, draft.
  23. PostScriptDOC`Adjacent Ramsey Theory', October 2, 1999, 3 pages, draft.
  24. PostScriptDOC`Quadratic Axioms', January 3, 2000, 9 pages, draft.
  25. PostScriptDOC`Geometry Axioms', January 8, 2000, 4 pages, draft.
  26. PostScriptDOC`Completeness of Intuitionistic Propositional Calculus', March 20, 2000, 5 pages, draft.
  27. `The Ackermann function in elementary algebraic geometry', October 21, 2000, 18 pages, draft.
  28. `Selection for Borel relations', September 20, 2001, 24 pages. See Publications.
  29. `Boolean relation theory notes', October 6, 2001, 20 pages, draft.
  30. `Metamathematics of comparability', October 7, 2001, 24 pages. See Publications.
  31. `Finite reverse mathematics', October 19, 2001, 28 pages, draft.
  32. `Maximal nonfinitely generated subalgebras', October 19, 2001, 16 pages. See Publications.
  33. `Metamathematics of Ulm theory', November 25, 2001, 33 pages, draft.
  34. `Primitive independence results', July 13, 2002, 23 pages. See Publications.
  35. `Three quantifier sentences', July 14, 2002, 33 pages. See Publications.
  36. `A Way Out', August 28, 2002, 49 pages. See Publications.
  37. `Equational Boolean relation theory', September 3, 2002, 25 pages, draft.
  38. `Philosophical Problems in Logic', Seminar notes at the Princeton Philosophy Department, September-December, 2002, 107 pages.
  39. `Philosophy of Mathematics', Seminar notes at the Princeton Philosophy Department, September-December, 2002, 71 pages.
  40. `Sentential Reflection', January 5, 2003, 3 pages, draft.
  41. `Restrictions and extensions', February 17, 2003, 3 pages, draft.
  42. `Elemental sentential reflection', March 3, 2003, 6 pages, draft.
  43. `Similar subclasses', March 11, 2003, 3 pages, draft.
  44. `Working with Nonstandard Models', July 31, 2003, 24 pages. See Publications.
  45. `The number of certain integral polynomials and nonrecursive sets of integers, part 2', February 22, 2004, 13 pages. See Publications.
  46. `Strict reverse mathematics', January 31, 2005, 24 pages, draft.
  47. 'Relational system theory', May 26, 2005, 12 pages, abstract.
  48. 'The inevitability of logical strength', May 31, 2005, 13 pages, draft.
  49. 'Pi01 Incompleteness: finite set equations', December 9, 2005, 6 pages, abstract. This supercedes the previous 11/30/05, 12/3/05, 12/4/05 abstracts.
  50. 'Pi01 Incompleteness: finite graph theory 1', January 30, 2006, 4 pages, abstract.
  51. 'Pi01, Pi00 Incompleteness: finite graph theory', February 25, 2006, 4 pages, abstract.
  52. (with J. Avigad), Combining decision procedures for the reals, 42 pages. See Publications.
  53. 'Concept Calculus', October 25, 2006, 42 pages, abstract. Supercedes August 7, 2006 version.
  54. 'Exotic Prefix Theory', September 14, 2006, 7 pages, abstract. Supercedes earlier version.
  55. 'What is o-minimality?', November 30, 2007, 19 pages, Supercedes earlier versions. See Publications.
  56. 'Formal statements of Goedel's second incompleteness theorem', January 14, 2007, 8 pages, abstract.
  57. 'New Borel Independence Results', May 7, 2007, 20 pages. Many proofs sketched.
  58. 'My Forty Years On His Shoulders', September 28, 2007, to appear in the proceedings of the Goedel Centenary meeting in Vienna, held in April, 2006, Horizons of Truth, 71 pages. Supercedes earlier versions.
  59. 'The Inevitability of Logical Strength: strict reverse mathematics', August 29, 2007, 65 pages, submited for publication. Supercedes earlier versions.
  60. The Nineteenth Annual Tarski Lectures. Interpretations of Set Theory in Discrete Mathematics and Informal Thinking. Delivered April 9,11,13, 2007 at the University of California at Berkeley. Manuscripts based on these lectures: 1. Interpretations, According to Tarski, May 24, 2007, 42 pages. Many proofs sketched. 2. Interpreting Set Theory in Discrete Mathematics: Boolean Relation Theory, May 24, 2007, 15 pages. Abstract. 3. Interpreting Set Theory in Ordinary Thinking: Concept Calculus, 12 pages. Abstract.

2. Lecture Notes

  1. `Invited lectures', January, 1997 to December 31, 2006, 16 pages.
  2. PostScriptDOC`The Formalization of Mathematics', February, 1997, 11 pages.
  3. PostScriptDOC`Ramsey Theory and Enormous Lower Bounds', April 5, 1997, 11 pages.
  4. `Transfer Principles in Set Theory', May 7, 1997, 21 pages.
  5. PostScriptDOC`Axiomatization of Set Theory by Extensionality, Separation, and Reducibility', October 5, 1997, 12 pages.
  6. PostScriptDOC`From Russell's Paradox to Higher Set Theory', October 10, 1997, 13 pages.
  7. PostScriptDOC`Finite Trees and the Necessary Use of Large Cardinals', January 10, 1998, 8 pages.
  8. PostScriptDOC`Unprovable Theorems in Discrete Mathematics', April 26, 1999, 10 pages.
  9. PostScriptDOC`A Theory of Strong Indiscernibles', April 28, 1999, 2 pages.
  10. PostScriptDOC`A Complete Theory of Everything: satisfiability in the universal domain', October 10, 1999, 15 pages.
  11. PostScriptDOC`Decreasing Chains of Algebraic Sets', November 12, 1999, 12 pages.
  12. PostScriptDOC`Discrete Independence Results', November 16, 1999, 13 pages.
  13. PostScriptDOC`The Mathematical Meaning of Mathematical Logic', April 21, 2000, 14 pages.
  14. PostScriptDOC`Does Mathematics Need New Axioms?', May 30, 2000, 13 pages.
  15. PostScriptDOC`A Logician Looks at Programming', May 31, 2000, 4 pages.
  16. PostScriptDOC`Foundations of Mathematics: past, present, and future', May 31, 2000, 8 pages.
  17. PostScriptDOC`Enormous Integers in Real Life', June 1, 2000, 11 pages.
  18. PostScriptDOC`New Axioms Panel', June 5, 2000, 8 pages.
  19. `Lecture notes on baby Boolean relation theory', October 3, 2001, 11 pages.
  20. `Does normal mathematics need new axioms?', October 26, 2001, 12 pages.
  21. `Lecture notes on term rewriting and computational complexity', November 7, 2001, 12 pages.
  22. `Lecture notes on enormous integers', November 22, 2001, 11 pages.
  23. `Issues in the foundations of mathematics', Godel Lecture, Association for Symbolic Logic, June 2, 2002, 14 pages.
  24. `Demonstrably necessary uses of abstraction', Hans Rademacher Lectures, University of Pennsylvania, September 17-20, 2002, 53 pages.
  25. `Adventures in the foundations of mathematics', VIGRE Lectures, Ohio State University, April 28-May 7, 2003, 64 pages.
  26. `Unprovable theorems', April 19, 2005, 11 pages.
  27. `Clay Millenium Problem: P = NP', Mathematics Colloquium, Ohio State University, October 20, 2005, 19 pages.
  28. 'What you cannot prove 1: before 2000', December 19, 2005, 18 pages.
  29. 'Forty years on his shoulders', Goedel Centenary, April 25, 2006, 9 pages.
  30. 'Remarks on Goedel Phenomena and the field of reals', Goedel Centenary, April 27, 2006, 6 pages.
  31. 'Remarks on the unknowable', Goedel Centenary, April 28, 2006, 6 pages.
  32. 'Adventures in the Verification of Mathematics', Computer Science Colloquium, Ohio State University, June 8, 2006, 19 pages.
  33. 'Search for Consequences', Logic Colloquium '06, August 9, 2006, Nijmegen, Netherlands, 15 pages.
  34. 'Goedel's Legacy in Mathematical Philosophy', Logic Colloquium '06', August 9, 2006, Nijmegen, Netherlands, 6 pages.
  35. 'Contemporary perspectives on Hilbert's second problem and the Goedel incompleteness theorems', AMS panel, New Orleans, January 6, 2007, 7 pages.
  36. 'Computer assisted certainty', Computer Science Lecture Series, University of Kentucky, January 25, 2007, 17 pages.
  37. 'Concept Calculus', Mathematical Methods in Philosophy, Banff, Canada, February 21, 2007, 9 pages.
  38. 'Limitations on our understanding of the behavior of simplified physical systems', Science, Reason, and Truth, San Marino, August 17-19, 2007, 26 pages.
  39. 'Concept Calculus', APA Panel on Logic in Philosophy, APA Eastern Division Annual Meeting, Baltimore Maryland, January 2, 2008, 17 pages. Supercedes 1/1/2008 and earlier versions.

3. Book Drafts

Expect that the various sections will be updated from time to time. All sections have the date 10/01/07 (except Appendix B by Francoise Point). Look for changes in the dates.

BOOLEAN RELATION THEORY AND THE INCOMPLETENESS PHENOMENA

PREFACE. 10/30/07. 2 pages.

INTRODUCTION. THE INCOMPLETENESS PHENOMENA. in preparation.

CHAPTER 1. INTRODUCTION TO BRT.
1.1. General Formulation. 10/0107. 19 pages.
1.2. Some BRT Settings. 10/01/07. 8 pages.
1.3. Complementation Theorems. 10/01/07. 15 pages.
1.4. Thin Set Theorems. 10/01/07. 15 pages.

CHAPTER 2. CLASSIFICATIONS.
2.1. Methodology. 10/01/07. 19 pages.
2.2. EBRT, IBRT in A,fA. 10/01/07. 10 pages.
2.3. EBRT, IBRT in A,fA,fU. 10/01/07. 24 pages.
2.4. EBRT in A,B,fA,fB,inclusion on (SD,INF). 10/01/07. 15 pages.
2.5. EBRT in A,B,fA,fB,inclusion on (ELG,INF). 10/01/07. 21 pages.
2.6. EBRT in A1,...,Ak,fA1,...,fAk,inclusion on (MF,INF). 10/01/07. 2 pages.
2.7. IBRT in A1,...,Ak,fA1,...,fAk,inclusion on (SD,INF), (ELG,INF), (MF,INF). 10/01/07. 12 pages.

CHAPTER 3. 6561 CASES OF EQUATIONAL BOOLEAN RELATION THEORY.
3.1. Preliminaries. 10/01/07. 30 pages.
3.2. Some Useful Lemmas. 10/01/07. 6 pages.
3.3. Single Clauses (duplicates). 10/01/07. 5 pages.
3.4. AAAA. 10/01/07. 3 pages.
3.5. AAAB. 10/01/07. 5 pages.
3.6. AABA. 10/01/07. 10 pages.
3.7. AABB. 10/01/07. 3 pages.
3.8. AABC. 10/01/07. 13 pages.
3.9. ABAB. 10/01/07. 4 pages.
3.10. ABAC. 10/01/07. 5 pages.
3.11. ABBA. 10/01/07. 3 pages.
3.12. ABBC. 10/01/07. 14 pages.
3.13. ACBC. 10/01/07. 6 pages.
3.14. Annotated Table of Representatives. 10/01/07. 26 pages.
3.15. Some Facts. 10/01/07. 4 pages.

CHAPTER 4. PROOF OF EXOTIC CASE.
4.1. Strongly Mahlo Cardinals of Finite Order. 10/01/07. 9 pages.
4.2. Proof using Strongly Mahlo Cardinals. 10/01/07. 22 pages.
4.3. Some Existential Sentences. 10/01/07. 10 pages.
4.4. Proof using 1-consistency. 10/01/07. 12 pages.

CHAPTER 5. INDEPENDENCE OF EXOTIC CASE.
5.1. Propositions C,D and Length 3 Towers. 10/01/07. 10 pages.
5.2. From Length 3 Towers to Length n Towers. 10/01/07. 17 pages.
5.3. Countable Nonstandard Models with Limited Indiscernibles. 10/01/07. 26 pages.
5.4. Limited Formulas, Limited Indiscernibles, x-definability, Normal Form. 10/01/07. 13 pages.
5.5. Comprehension, Indiscernibles. 10/01/07. 8 pages.
5.6. Pi01 Correct Internal Arithmetic, Simplification. 10/01/07. 20 pages.
5.7. Transfinite Induction, Comprehension, Indiscernibles, Infinity, Pi01 Correctness. 10/01/07. 26 pages.
5.8. ZFC + V = L, Indiscernibles, and Pi01 Correct Arithmetic. 10/01/07. 27 pages.
5.9. ZFC + V = L + {(therexists kappa)(kappa is strongly k-Mahlo)}_k + TR(Pi01,L), and 1-Con(SMAH). 10/01/07. 13 pages.

CHAPTER 6. FURTHER RESULTS.

6.1. Propositions D,E,F. 10/01/07. 15 pages.
6.2. Effectivity. 10/01/07. 24 pages.
6.3. A Refutation. 10/01/07. 3 pages

REFERENCES. 10/01/07. 6 pages.

APPENDIX A. 10/01/07. 2 pages.

APPENDIX B. 8/27/07. By Francoise Point. On the Expansion (N;+;2^x) of Presburger Arithmetic. 15 pages.

ENTIRE BOOK, SINGLE FILE. BOOLEAN RELATION THEORY AND THE INCOMPLETENESS PHENOMENA. 10/30/07 version. Same as 10/01/07 version with Preface added. 568 pages without Appendix B. See above for Appendix B by Francoise Point.


4. FOM e-mail List

The FOM e-mail list is an automated e-mail list devoted to lively discussion of the foundations of mathematics. I founded FOM (jointly with S. Simpson) in late 1997.

Information about the FOM e-mail list can be obtained at http://www.cs.nyu.edu/mailman/listinfo/fom

The FOM archives are located at http://www.cs.nyu.edu/pipermail/fom.  My postings to the FOM e-mail list can be obtained there.


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