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I,II, abstracts, J. of Symbolic Logic, Vol. 41, No. 2 (1976), pp. 557-559, as well as the following widely distributed unpublished manuscripts.The Analysis Of Mathematical Texts, And Their Calibration In Terms Of Intrinsic Strength I, April 3, 1975, 7 pages. hisAnalMathTxtI The Analysis Of Mathematical Texts, And Their Calibration In Terms Of Intrinsic Strength II, April 8, 1975, 5 pages. hisAnalMathTxtII The Analysis Of Mathematical Texts, And Their Calibration In Terms Of Intrinsic Strength III , May 19, 1975, 26 pages. histAnalMathTxtIII The Analysis Of Mathematical Texts, And Their Calibration In Terms Of Intrinsic Strength IV, August 15, 1975, 32 pages. histAnalMathTxtIV The Logical Strength Of Mathematical Statements, October 15, 1975, 1 page. hisLogStr The Logical Strength Of Mathematical Statements I’, August, 1976, 20 pages. hisLogStr1 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. Combinatorial set theoretic statements of great logical strength, 1995, 4 pages, abstract. CombPrinc95

3. Some Decision Problems Related to Hilbert’s Tenth Problem, January 15, 1996, 2 pages, abstract. Hilbert10th1%5B1%5D.15.96

4. Extremely large cardinals in the rationals, August 1996, 4 pages, abstract. ExtLC96

5. Countable model theory and large cardinals, September 1996, 3 pages, abstract. CountModel96

6. On Expansions of o-Minimal Structures, November 23, 1996, 4 pages, draft. O-minExp96

7. The Formalization of Mathematics, May 21, 1997, 11 pages, draft. FormMath97

8. Some Historical Perspectives on Certain Incompleteness Phenomena, May 21, 1997, 5 pages, draft. HIstPersInc97

9. Transfer Principles in Set Theory, May 21, 1997, 44 pages, draft. transprin_5_21_97

10. The interpretation of set theory in pure predication theory, August 20, 1997, 64 pages, draft. IntSetPred97

11. Applications of Large Cardinals to Graph Theory, October 23, 1997, 36 pages, draft. AppLCtoGT97

12. The interpretation of set theory in mathematical predication theory, October 25, 1997, 10 pages, draft. IntSetMPT97

13. The axiomatization of set theory by extensionality, separation, and reducibility, October 28, 1997, 71 pages, draft. AxSetExt97

14. Applications of Large Cardinals to Borel Functions, November 26, 1997, 3 pages, abstract. AppLCbf97

15. Finite Trees and the Necessary Use of Large Cardinals, March 22, 1998, 58 pages, draft. FinTreNec98

16. Subtle Cardinals and Linear Orderings, April 1998, 52 pages, See Publications. subtlecardinals

17. Long Finite Sequences, October 8, 1998, 50 pages. See Publications. LongFinSeq98

18. Interpretability and Definability in Expansions, with Chris Miller, November, 1998, 5 pages, abstract. IntDefExp98

19. Some Decision Problems of Enormous Complexity, May 6, 1999, 10 pages. See Publications. somedecproenocom5_6_99

20. A Consistency Proof for Elementary Algebra and Geometry, August 23, 1999, 38 pages, draft. ConsRCF8%5B1%5D.23.99

21. Borel and Baire reducibility, September 2, 1999, 11 pages. See Publications. BorelBaireReduc090299

22. Conservation, September 24, 1999, 7 pages, draft. Conservation9=24=99

23. Adjacent Ramsey Theory, October 2, 1999, 3 pages, draft. See #65 below. AdjRamTh10%5B1%5D.2.99

24. Quadratic Axioms, January 3, 2000, 9 pages, draft. QuadAxiom=1%5B1%5D.6.00

25. Geometry Axioms, January 8, 2000, 4 pages, draft. GeomAxioms1%5B1%5D.8.00

26. Completeness of Intuitionistic Propositional Calculus, March 20, 2000, 5 pages, draft. CompltIntuiLogic3%5B1%5D.20.00

27. The Ackermann function in elementary algebraic geometry, October 21, 2000, 18 pages, draft. AckAlgGeom102100

28. Selection for Borel relations, September 20, 2001, 24 pages. See Publications. BorelSelection093001

28. Metamathematics of comparability, October 7, 2001, 24 pages. See Publications. MetaComp100701

29. Boolean relation theory notes, October 6, 2001, 20 pages, draft. BoolRelnThyNotes100601

30. Finite reverse mathematics, October 19, 2001, 28 pages, draft. FiniteRevMath101901

31. Maximal nonfinitely generated subalgebras, October 19, 2001, 16 pages. See Publications. MaxSubalg101901

32. Metamathematics of Ulm theory, November 26, 2001, 33 pages, draft. MetaUlmThy112501

33. Primitive independence results, July 13, 2002, 23 pages. See Publications. PrimitiveIndResults071302

34. Three quantifier sentences, July 14, 2002, 33 pages. See Publications. three%20quantifiers071402

35. A Way Out, August 28, 2002, 49 pages. See Publications. AwAYOUT

36. Equational Boolean relation theory, September 3, 2002, 25 pages, draft. EBRT090302

37. Philosophical Problems in Logic, Seminar notes at the Princeton Philosophy Department, September-December, 2002, 107 pages. Princeton532

38. Philosophy of Mathematics, Seminar notes at the Princeton Philosophy Department, September-December, 2002, 71 pages. Princeton536

39. Sentential Reflection, January 5, 2003, 3 pages, draft. SentRfn010503

40. Restrictions and extensions, February 17, 2003, 3 pages, draft. ResExt021703

41. Elemental sentential reflection, March 3, 2003, 6 pages, draft. ElSentRef030303

42. Similar subclasses, March 11, 2003, 3 pages, draft. SimSubc031103

43. Working with Nonstandard Models, July 31, 2003, 24 pages. See Publications. Balt073103

44. The number of certain integral polynomials and nonrecursive sets of integers, part 2, February 22, 2004, 13 pages. See Publications. NatNon022204

45. Strict reverse mathematics, January 31, 2005, 24 pages, draft. StrictRM012305

46. Relational system theory, May 26, 2005, 24 pages, draft. TernaryAbstract052605

47. The inevitability of logical strength, May 31, 2005, 13 pages, draft. nevLogStr082907

48. Pi01 Incompleteness: finite set equations, December 9, 2005, 6 pages, abstract. Pi01120905

49. Pi01 Incompleteness: finite graph theory 1, January 30, 2006, 4 pages, abstract. Pi01013006

50. Pi01, Pi00 Incompleteness: finite graph theory, February 25, 2006, 4 pages, abstract. Pi01013006

51. (with J. Avigad), Combining decision procedures for the reals, 42 pages. See Publications. JointAvigadAppear

52. Concept Calculus, October 25, 2006, 42 pages, abstract. Supercedes August 7, 2006 version. ConceptCalc102506,pdf

53. Exotic Prefix Theory, September 14, 2006, 7 pages, abstract. Supercedes earlier version. ExoticPrefix091406

54. What is o-minimality?, November 30, 2007, 19 pages. See Publications. o-minimality113007

55. Formal statements of Goedel’s second incompleteness theorem, January 14, 2007, 8 pages, abstract. FormalGodel2nd020107

56. New Borel Independence Results, May 7, 2007, 20 pages. Many proofs sketched. New%20Borel050707

57. My Forty Years On His Shoulders, October 31, 2009, to appear in the proceddings of the Goedel Centenary meeting in Vienna, held in April 2006, Horizons of Truth, 69 pages. Supercedes earlier versions. Fortyyears111909

58. The Inevitability of Logical Strength: strict reverse mathematics, 65 pages. See Publications. InevLogStr082907

59. 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. Tarski1,052407 2. Interpreting Set Theory in Discrete Mathematics: Boolean Relation Theory, May 24, 2007, 15 pages. Abstract. Tarski2,052407 3. Interpreting Set Theory in Ordinary Thinking: Concept Calculus, 12 pages. Abstract. Tarski3,052407

60. (with S. Kieffer, J. Avigad), A language for mathematical knowledge management, 16 pages, August 2008. To appear in special issue of Studies in Logic, Grammar and Rhetoric, 2008. http://www.andrew.cmu.edu/user/avigad/Papers/mkm/index.html See Publications. LangMathKnow08

61. Limitations on our Understanding of the Behavior of Simplified Physical Systems, October 30, 2008, 26 pages. See publications. SanMarino111608

62. Concept Calculus: Much Better Than, October 31, 2009, 48 pages. To appear in New Frontiers in Research on Infinity, ed. Michael Heller and W. Hugh Woodin, Cambridge University Press, 2010. (Initial paper on Concept Calculus). See Publications. ConcCalcInf103109

63. Boolean Relation Theory and More. This is actually a sketchy treatment of The Upper Shift Fixed Point Theorem, given at the MIT Logic Seminar, November 3, 2009. 10 pages. MITLogicSem110309

64. Decision Problems in Euclidean Geometry. This is an advanced draft nearing completion for publication. August 29, 2010, 33 pages. This version (supercedes August 23, 2010 version) is based on Integral Domains of Characteristic Zero, and Ordered Rings. Hilb10Geom082910

65. Adjacent Ramsey Theory. Draft. Key proofs are given. August 29, 2010, 17 pages. PA incomp082910

66. Kernel Structure Theory. Abstract. Independent Pi01 sentences. August 29, 2010, 8 pages. KernStruThm100910

67. Equational Representations. This is an advanced draft nearing completion for publication. September 23, 2010. 17 pages. EquatReps092310

68. Finite Phase Transitions. Draft. September 26, 2010. 9 pages. NOTE: This manuscript initiates a new line of research on phase transitions, but Royce Peng has pointed out that it has errors in the details. He is in communication concerning stronger claims. phasetrans092610′

69. The Upper Shift Kernel Theorems. Draft. October 9, 2010. 7 pages. Supersedes earlier versions. KernStruThm100910

70. Invariant Maximal Cliques and Incompleteness, October 7, 2011 version, 132 pages. Supersedes original October 4, 2011, and October 6, 2011 versions. Corrected several inaccurate statements, improved exposition. MaxClique100711

71. Embedded Maximal Cliques and Incompleteness. Extended Abstract. 18 pages. May 20, 2013. Reesults supersede 71. Largely supersedes 72, November 6, 2012, but some matters from 72 are not addressed in this new version, and so we are keeping 72. EmdMaxNotes052013

72. Concept Calculus: universes. October 2, 2012, 33 pages, submitted for publication. Supersedes September 30, 2012 version. CocCalc-univ100212

73. Unique Undefinable Elements. January 9, 2013, 33 pages, submitted for publication. Supersedes November 27, 2012 version. UniqueUndElt010913

74. A Divine Consistency Proof for Mathematics. December 25, 2012, 70 pages. Submitted for publication. ConMathThe122512

75. Sum Base Towers, Invariant Maximal Continuations, and Concrete Incompleteness. Extended abstract. 48 pages. August 2, 2013. 72 is incorporated as a small part. Some matters in 71 not covered here, so we are keeping 72. MaxIncAbs080213

76. Invariant Maximality and Incompleteness, 37 pages, April 30, 2014. Supersedes January 8, 2014 version. This has appeared 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.** CMI051114w**

77. Order Theoretic Equations, Maximality, and Incompleteness, 17 pages, June 7, 2014. Extended abstract. Largely supersedes 77, but some content in 77 is not incorporated here, and so we are keeping 77. OrdTheEq060714

78. Order Invariant Graphs and Finite Incompleteness, 4 pages, June 22, 2014. Extended abstract. FIiniteSeqInc062214a We have independence from ZFC for small k,n. We are targeting k = n = 4.

79. Countable Model Theory and Incompleteness, 3 pages, June 27, 2014. Extended abstract. Minor revision of previous version. CounModInc062714

80. Order Invariant graphs and Incompleteness, 3 pages, July 1, 2014. Extended abstract. The invariance condition is now stated directly in familiar terms. OrdInvGra070114

81. Order Invariant Relations and Incompleteness, 14 pages, September 2a, 2014. Extended abstract. Includes an explicitly Pi01 sentence equivalent to Con(HUGE). This supersedes 7/19/14, 8/5/14, 8/6/14, 8/16/14, 8/18, 8/22, 9/1, 9/2. OrdInvInc090214a

82. Testing the Consistency of Mathematics, 7 pages, July 23, 2014. Extended abstract. TestCons072314

83. Flat Mental Pictures, September 5, 2014, 5 pages. Extended abstract. This supersedes 8/11/14, 8/18/14, 8/19/14, 8/19a/14, and Exotic Self Embedding Axiomatization, 8/2014. . Mental090514

84. Universal Properties and Incompleteness, October 4, 2014, 23 pages. Obsoletes 9/15/14, 9/15a/14, 9/15b/14, 9/26/14. Extended abstract. Supersedes 82, which is largely incorporated. UnivPropInc100414.

85. Conservative Growth: A Unified Approach to Logical Strength, October 12, 2014, 15 pages. ConsGwth101214

86. Mathematically Natural Concrete Incompleteness, June 21, 2015, 60 pages. Supersedes 1/15/15, 1/10/15, 1/13/15 versions. Putnam062115pdf

87. Impossible Counting, June 2, 2015, 10 pages, draft. Supersedes the May 26, 2015 version. All changes after Theorem 3.2. Count060215

88. On the Principle of Consistent Truth, June 20, 2016, 13 pages. ConsTruth062016

89. Monadic Partial Functions with Objects and Sets, June 1 – August 14, 2016, 10 pages, preliminary report. DecPartFcn071416

90. Complete Dense Linear Orderings, December 8, 2016, 25 pages. charreals120816

91. Expanding Mind Theory, November 18, 2016, 17 pages. expandmind111816

92. Concrete Mathematical Incompleteness: Basic Emulation Theory, July 1, 2017, 78 pages. Submitted to the Putnam Volume, ed. Cook, Hellman. Revised October 10, 2017. (Has now appeared in 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.) PutnamVol101017-2i8ths2

93. Digital Display Expansion, July 7, 2017, 18 pages. I wanted to get this out quickly. An advanced draft, but still needs more proofreading and polishing. Will update soon. PS: We have already changed the terminology from “expansion” to “enrichment”. I.e., we refer to “fully enrichable” instead of “fully expandable”. Pixels070717-142mnko

94. Impossible Counting, September 2, 2017, 76 pages. There is an earlier item above, 87. Impossible Counting, 6/2/15, which is mostly an abstract, and contains some still interesting material not present in this 94. It does contain a claim that amounts to “theta(2,12) is impossible to count in ZFC and strong large cardinal extensions. Here we give complete proofs for the Impossible Counting of theta(4,7). We retract this earlier claim. ImpossibleCnt090217-2a5ewhm

95. Integer Inner Product and Algorithmic Unsolvability, 23 pages, October 13, 2017. Revised October 26, 2017, 23 pages. See also the earlier work above, 64. Decision Problems in Euclidean Geometry, August 29, 2010. InnerProducts102617-2bpu9hx

96. x = +-yz +-1 and Algorithmic Unsolvability, 6 pages, October 15, 2017. QuadEqns101517-1xhh83d

97. Integral Piecewise Quadratic F:Z5 into Z and Algorithmic Unsolvability, 6 pages, October 26, 2017. QuadraticSets102617-23599rv

98. Concrete Mathematical Incompleteness Status 2/8/18, 3 pages, February 8, 2018. CMIstatus020818-ucpy18

99. Concrete Mathematical Incompleteness Status 3/6/18, 14 pages, March 6, 2018. CMIstatus030618-1cwlagj

100. Order Emulation Theory Status 3/27/18, 10 pages, Supersedes the March 25, 2018 version. CMIcomOrdEm032718-qi0eip

101. Adventures in Piano – 3/28/18, 8 pages. PianoAdv032818-1tufkhl

102. Explicitly Pi01 Status – 4/20/18. 6 pages. Presents state of the art explicitly Pi01 statements requiring large cardinals as of 4/20/18. Obsoletes all earlier versions that were previously numbered 102, 103. CMI_ExplicPi01042018-2ilxsqy

103. Upper Image Stability, 7 pages. April 22, 2018. Greatly improved thematically and conceptually from 102, 4/20/18, though 102 is worth keeping. CMIupperImage042218-2d74fav

104. Upper Image Stability, 6 pages. April 26, 2018. An essentially cosmetic improvement on 103, but at this very advanced stage such cosmetic improvements can make a very large difference particularly for casual readers. We keep 103. CMIupperImage042618-11k6yex

105. Concrete Mathematical Incompleteness Highlights 4/29/18, 7 pages, April 29, 2018. State of the art incorporating 104. Some discussion of the Principle of Invariant Maximality. CMIstatus042918-1qbagkz This was intended to close Phase 1 CMI (at least temporarily), but there has been considerable further developments. We are preparing an update to these Highlights that takes into account these subsequent developments. In the meantime, we have spotted the need for a correction to section 5 here, June 12, 2018: CMIcorrect061218-1323h9c

106. Tangible Mathematical Incompleteness of ZFC, July 14c, 2018, 25 pages. Improved treatment of fully stable. Supersedes July 14b, 2018 version. NEWS! S contained in Q[0,k]^k is stable if and only if for all p_1,…,p_i < i, (p_1,…,p_i,i,…,k-1) in S iff (p_1,…,p_i,i+1,…,k) in S. This seems to be the simplest notion of stability that seems to give independence from ZFC – being checked. There is a natural galaxy of such conditions, culminating in the strongest one called full stability. This is invariance under critical equivalence, which has a new definition using strictly increasing enumerations. x,y in Q[0,k]^k are critically equivalent if and only if x,y are order equivalent, and their strictly increasing enumerations both lie in {1,…,k} at or past any position at which they differ. EveryMath071418c-zp65bc

107. Computational Confirmation of Consistency, 8 pages, July 3, 2018. Supersedes Order Constrained Sequential Set Constructions, 6/27/18. SeqConsGraphs070318-1rwj9yv

108. Tangible Mathematical Incompleteness of ZFC, August 16, 2018, 68 pages. Supersedes 106, but we are retaining 106 for the historical record. This is a fairly polished advanced draft, but we will be making some additions and refinements in due course. This is the second in the trilogy consisting of #92, this #108, and the projected “reversal” paper deriving the consistency of some large cardinals (Con(SRP)) from the lead statements in Emulation Theory. All of this will be reassembled and extended along with BRT (Boolean Relation Theory) into a book with the tentative title of Concrete Mathematical Incompleteness. ALERT 8/17/18: We are now using a multiple form of the default stability in #108, which reads: for all p_1,…,p_i < i, (p_1,…,p_i,i,…,k-1) in S iff (p_1,…,p_i,i+1,…,k) in S. This change in the default does not materially impact #108 in any way. This #108 will be updated as things develop. EveryMath081618-24vxu2b

109. Tangible Incompleteness Interim Report October 12, 2019, 19 pages. Totally self contained, should be read independently of 108. Will be combined with 108 later. TangiibleIInerim101219

110. NEW SUPERSEDING. Tangible Incompleteness Interim Report March 31, 2020, 21 pages. Major overhaul. Major upgrade of January version. First priority now is to back up all claims in this abstract. Fully compatible with earlier versions, but with major expositional overhaul. Buzzword now INVARIANT MAXIMALITY. This appears to be the culmination of 53 years of effort to find Tangible Incompleteness in the extremely elemental combinatorial level. One major follow through will be to incorporate less elemental mathematical structures. Here we only use countable dense linear orderings and also finite intervals in the integers. TangIncomp033120

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