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Independent sentences of mathematical character

  • Duccio Pianigiani

The reception of incompleteness results within the mathematical community was very slow. One reason for the large underestimation of gödelians’ results in part of the community of mathematicians, which helped to brake their assimilation, was linked to the metamathematical character of the statement “I am not provable” used in the constructive proof. The perceived distance from the concrete mathematical work is perhaps behind the most striking case of the silence in this regard: that of the the French group ‘Bourbaki’ of formalists mathematicians. For this reason, in an attempt to address these objections, some scholars have devoted themselves to the search for independent statements of mathematical content, with proofs that did not rely on the typical Gödelian toolbox.

  • Keywords:
  • Paris-Harrington theorem,
  • Ramsey theorems,
  • Hydra Game,
  • Goodstein sequences,
+ Show More

Duccio Pianigiani

University of Siena, Italy - ORCID: 0000-0001-9441-7226

  1. Arai, Toshiyasu. 2020. Ordinal Analysis with an Introduction to Proof Theory. Berlin: Springer. DOI: 10.1007/978-981-15-6459-8
  2. Barwise, John and Jerome Keisler ed. 1989. Handbook of Mathematical Logic. Amsterdam: North Holland.
  3. Beklemishev, Lev. 2006. “The Worm principle”. In Z. Chatzidakis, P. Koepke, W. Pohlers (editors) Logic Colloquium ’02 75-95. Cambridge: Cambridge University Press. DOI: 10.1017/9781316755723.005
  4. Bourbaki. 1950. “The Architecture of Mathematics”. The American Mathematical Monthly 57 (4): 221-232. DOI: 10.1080/00029890.1950.11999523
  5. Buchholz, Wilfried and Wainer, Stan. 1987. “Provably computable functions and the fast growing hierarchy”. In S. Simpson (editor) Logic and Combinatorics. Contemp. Math 65 179-98. Providence: American Mathematical Society. DOI: 10.1090/conm/065
  6. Buchholz, Wilfried. 1987. “An independence result for (Π 11 − CA) + BI” . Annals of Pure and Appied Logic 33: 131-155. DOI: 10.1016/0168-0072(87)90078-9
  7. Carlucci, Lorenzo. 2003. “A new proof-theoretic proof of the independence of Kirby-Paris’ Hydra Theorem.” Theoretical Computer Science 300: 365-378. DOI: 10.1016/S0304-3975(02)00332-8
  8. Carlucci, Lorenzo. 2005. “Worms, gaps, and hydras”. Mathematical Logic Quarterly 51 (4): 342-350. DOI: 10.1002/malq.200410035
  9. Cichon, Adam. 1983. “A short proof of two recently discovered independence results using recursion theoretic methods”. Proceedings of the American Mathematical Society 87 (4): 704-706
  10. Dieudonné, Jean. 1987. Pour l’honneur de l’esprit humain. Paris: Hachette Littérature.
  11. Endrullis, Jörg and Klop, Jan and Overbeek, Roy. 2021. “Star Games and Hydras.”Logical Methods in Computer Science 17 (2):1-32. DOI: 10.23638/LMCS-17(2:20)2021
  12. Gallier, Jean H. 1991. “What’s so special about Kruskal’s theorem and the ordinal Γ 0 ? A survey of some results in proof theory” . Annals of Pure and Applied Logic 53 (3): 199-260. DOI: 10.1016/0168-0072(91)90022-E
  13. Goodstein, Reuben. 1944. “On the restricted ordinal theorem” . The Journal of Symbolic Logic 9: 33-41. DOI: 10.2307/2268019
  14. Gentzen, Gerhard. 1936. “Die Widerspruchfreiheit der reinen Zahlentheorie” Mathematische Annalen 112: 493-565 (reprinted in M. E. Szabo (editor), Collected papers of Gerhard Gentzen 1969. Amsterdam: North Holland 132-213).
  15. Grattan-Guinness, Ivor. 1979. “In Memoriam Kurt Gödel: his 1931 correspondence with Zermelo on his incompletability theorem” . Historia Math. 6: 294-304. DOI: 10.1007/BF03023829
  16. Grattan-Guinness, Ivor. 2011. “The Reception of Gödel’s 1931 Incompletability Theorems by Mathematicians, and Some Logicians, to the Early 1960s ”. In M. Baaz (editor) Kurt Gödel and the Foundations of Mathematics 57-74. Cambridge: Cambridge University Press. DOI: 10.1017/CBO9780511974236
  17. Hájek, Peter and Pudlák, Pavel. 1993. Metamathematics of first order arithmetic. Berlin: Springer.
  18. Hamano, Masahiro and Okada, Mitsuhiro. 1998. “A direct independence proof of Buchholz’s Hydra Game on finite labeled trees” . Archive for Mathematical Logic 37: 67–89. (1998). DOI: 10.1007/s001530050084
  19. Ketonen, Jussi and Solovay, Robert. 1981. “Rapidly Growing Ramsey Functions”. Annals of Mathematics 113 (2): 267-314. DOI: 10.1007/978-3-319-01315-2_8
  20. Kirby, Laurie and Paris, Jeff. 1982. “Accessible Independence Results for Peano Arithmetic”. Bulletin of the London Mathematical Society 14 (4): 285-293. DOI: 10.1112/blms/14.4.285
  21. Kreisel, Georg. 1952. “On the Interpretation of Non-Finitist Proofs: Part II. Interpretation of Number Theory. Applications” . The Journal of Symbolic Logic 17 (1): 43-58.
  22. Kripke, Saul A. 2021. “Mathematical Incompleteness Results in First-Order Peano Arithmetic: A Revisionist View of the Early History” . History and Philosophy of Logic 43 (2): 175-182. DOI: 10.1080/01445340.2021.1976052
  23. Macintyre, Angus. 2011. “The Impact of Gödel’s Incompleteness Theorems on Mathematics” . In Kurt Gödel and the Foundations of Mathematics, edited by M. Baaz 3-75. Cambridge: Cambridge University Press. DOI: 10.1080/01445340.2021.1976052
  24. Marker, David. 2002. Model Theory: An Introduction. Berlin: Springer. DOI: 10.1007/978-0-387-21787-7
  25. Paris, Jeff B. 1987. “Some independence results for Peano arithmetic.” The Journal of Symbolic Logic 43 (4): 725-731.
  26. Paris, Jeff and Harrington, Leo. 1977. “A mathematical incompleteness in Peano Arithmetic”. In Handbook of Mathematical Logic, edited by J. Barwise and : Keisler 1133-1142. Amsterdam: North-Holland.
  27. Rathjen, Michael. 2006. “The art of ordinal analysis.” In M. Sanz Solé, J. Soria de Diego, J. L. Varona Malumbres, J. Verdera (editors) Proceedings of the International Congress of Mathematicians 2 45-70. Berlin: EMS Press. DOI: 10.4171/022-2/3
  28. Rathjen, Michael. 2015. “Goodstein’s theorem revisited”. In R. Kahle and M. Rathjen (editors) Gentzen’s Centenary: The Quest for Consistency: 229-242. Berlin: Springer. DOI: 10.1007/978-3-319-10103-3_1. 3
  29. Simpson, Stephen ed. 1987. Logic and combinatorics, Proceedings of the AMS-IMS-SIAM joint summer research conference held August 4-10, 1985. Contemporary Mathematics, volume 65. Providence: American Mathematical Society. DOI: 10.2307/2275387
  30. Simpson, Stephen G. 1999. Subsystems of Second Order Arithmetic. Berlin: Springer. DOI: 10.1017/CBO9780511581007
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  • Publication Year: 2025
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Chapter Information

Chapter Title

Independent sentences of mathematical character

Authors

Duccio Pianigiani

Language

English

DOI

10.36253/979-12-215-0778-2.11

Peer Reviewed

Publication Year

2025

Copyright Information

© 2025 Author(s)

Content License

CC BY-SA 4.0

Metadata License

CC0 1.0

Bibliographic Information

Book Title

Lectures in Proof Theory and Complexity

Authors

Duccio Pianigiani

Peer Reviewed

Publication Year

2025

Copyright Information

© 2025 Author(s)

Content License

CC BY-SA 4.0

Metadata License

CC0 1.0

Publisher Name

Firenze University Press, USiena Press

DOI

10.36253/979-12-215-0778-2

eISBN (pdf)

979-12-215-0778-2

eISBN (xml)

979-12-215-0779-9

Series Title

UNIverSI. Ricerca e Didattica all’Università di Siena

Series ISSN

3035-5915

Series E-ISSN

3035-5931

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