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, https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1974.1/ICM1974.1.ocr.pdf, and Subsystems of Second Order Arithmetic with Restricted Induction I,II, abstracts, J. of Symbolic Logic, Vol. 41, No. 2 (1976), pp. 557-559, RMAbsracts1976 as well as the following widely distributed unpublished manuscripts.The Analysis Of Mathematical Texts, And Their Calibration In Terms Of Intrinsic Strength I, April 3, 1975, 7 pages.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.
For a detailed history of Reverse Mathematics and the less developed Strict Reverse Mathematics, see manuscript 116 on this page.
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. ALERT. BRAND NEW. Tangible Incompleteness Interim Report August 9, 2020, 31 pages. This is updated from July 16, July 30, 2020. Upgrade is mainly new improved finite forms, explicitly Pi01. New Major Overhaul of March version. First priority now is to back up all claims in this abstract. Major content overhaul and major expositional overhaul. 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. TangibleIncomReport080920
111. Simplified Axioms for Class Theory. September 15, 2020, 22 pages. NBGAxioms091520
112. Mappings of Symmetric Semigroups, February 2, 2021, 53 pages. Revised February 3, 2021, 55 pages. Some minor reworking the main proof at a messy spot. Also put in emphasis that all relevant mappings preserve the identity so I can be viewed as working exclusively with monoids. Revised February 6, 2021, 59 pages. Added section proving that all absolutely preserving mappings on all symmetric semigroups are automorphisms. SymmetricSemigroups020621
113. Regular Polygons and Tangible Incompleteness, April 25, 2021, 4 pages. Announcement. Research inspired by second Gent Talk, see Downloadable Lecture Notes. TangiblePoly042521
114. Integer and Word Sequences: Tangible Incompleteness, April 25, 2021. Announcement. Research inspired by second Gent Talk, see Downloadable Lecture Notes. TangIntWord042521,pdf
115. Uniform Descent and Tangible Incompleteness, announcement. May 9, 2021, 2 pages. TangIncDescent050921
116. The Emergence of (Strict) Revese Mathematics, December 29, 2021, 110 pages. RMfoundingETF122921a
117. Stable Maximality for Gifted Youth: Emulator Theory. Seven lectures at the 2022 Summer Ross Program. Lecture 1, Logical Reasoning, June 27, 2022, RossProg#1,072322pdf Lecture 2, Finite Set Theory, June 29, ROSSprog#2,072322 Lecture 3, Infinite Sets and ZFC, July 1, ROSSprog#3,072322 Lecture 4, Order Equivalence, Emulators, and Stability, July 4, ROSSprog#4,072322 Lecture 5, Stability of Emulators of Two Pairs, July 6, ROSSprogram#5,0723221 Lecture 6, Stability of Emulators of Three Pairs, July 6 ROSSprog#6,072222* Lecture 7, Final Talk, July 22, 2022, ROSSprog#7,072322
118. String Replacement Systems, September 18, 2022, 29 pages. Supersedes May 6, 2022 version. StringReplace091822
119. Adventures in Godel Incompleteness, September 18, 2023, 20 pages. IncompChinese0918233
120. A Divine Consistency Proof for Mathematics, September 19, 2022, 73 pages. This is an update of 74 above, with minor corrections. ConMathThe091922
Corrections flagged by Saeed Salehi, 6/26/24. DivinityCorr062624
121. Tangible Incompleteness Abstract, Invariant Maximality, June 18, 2023, 8 pages. Superseded June 26, 2023, 10 pages. Corrections and expansion. NOTE: Remove Propositions L and P from the manuscript. State of the art for SRP and HUGE, both infinite and finite in Q,N. TangIncUpdate062623
122. Finite Invariant Choice, August 8, 2024, Chapter 5 of the Invariant Maximality, Choice, and Games book under preparation. 9 pages. TangibleFin080824
This is being superseded and will be replaced by Finite Invariant Maximality, Chapter 5.
123. Polynomial and Quadratic Incompleteness, March 26, 2024, 5 pages.PAIncomp032624
124. Shift, Lower Shift, Interval Shift Invariance, April 21, 2024. Chapter 2 of the Invariant Maximality, Choice, and Games book under preparation. 36 pages. TangIncInv042124.per
Will be updated soon.
125. Polynomial Incompleteness, September 13, 2024, 6 pages. I prefer this to 123, but 123 needs to stay. PAIncomp091324
126. Invariant Maximality Reversals, Chapter 4 of the Invariant Maximality, Choice, and Games book under preparation. This is sections 4.1-4.3. Will be updated as the reversal lectures are delivered. November 23, 2024. TangRevSec1-3,112324