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Abstract views of incompleteness

  • Duccio Pianigiani

The aim of this section is to account for Gödel’s original intuition with the means of modern computability theory, showing the different complexity of the theorems of a theory of formal arithmetic with certain properties, and of the set of true propositions of the language of this theory. For this purpose, we refine the notion of computably enumerable set through the concept of creative set, introduced by Emil Post.

  • Keywords:
  • Recutsive functions,
  • recursive enumerability,
  • crative sets,
  • productive sets,
  • incompleteness,
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Duccio Pianigiani

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

  1. Amidei, Jacopo and Pianigiani, Duccio and San Mauro Luca and Simi, Giulia and Sorbi, Andrea. 2016. “Trial and error Mathematics I.” The Review of Symbolic Logic 9(2): 299-324. DOI: 10.1017/S1755020315000404
  2. Amidei Jacopo and Pianigiani, Duccio and San Mauro Luca and Sorbi, Andrea. 2016. “Trial and error mathematics II: dialectical sets and quasi-dialectical sets, their degrees, and their distribution within the class of limit sets.” The Review of Symbolic Logic 9(4): 810-835.
  3. Amidei Jacopo and Andrews, Uri and Pianigiani, Duccio and San Mauro Luca and Sorbi, Andrea. 2019. “Trial and error mathematics III: Dialectical systems and completions of theories.”Journal of Logic and Computation, 29 (1): 157-184 DOI: 10.1093/logcom/exy033
  4. Bernardi, Claudio. 1981. “On the relation provable equivalence and on partitions in effectively inseparable sets” . Studia Logica 40: 29-37. DOI: 10.1007/BF01837553
  5. Boykan Pour-El, Marian. 1968. “Effectively extensible theories” . The Journal of Symbolic Logic 33 (1): 56-68. DOI: 10.2307/2270052
  6. Buss, Samuel R. 1998. “First-order proof theory of arithmetic.” In Handbook of Proof Theory, edited by S. Buss, 79-147. Amsterdam: Elsevier.
  7. Cheng, Yong. 2023. “There Are No Minimal Effectively Inseparable Theories ” Notre Dame Journal of Formal Logic 64 (4): 425-439. DOI: 10.1215/00294527-2023-0017
  8. Cooper, Barry S. 2003. Computability Theory. Boca Raton: Chapman and Hall/ CRC Press DOI: 10.1201/9781315275789
  9. Cooper, Barry S. 2004. “The Incomputable Alan Turing” . In Proceedings of the 2004 inter- national conference on Alan Mathison Turing: a celebration of his life and achievements (Turing’04), edited by J. Delve and J. Paris, 1-17. Swindon: BCS Learning and Development Ltd.
  10. Copeland, Jack. The Church-Turing Thesis. alanturing.net
  11. Copeland, B. Jack and Shagrir, Oron. 2013. “Turing versus Gödel on Computability and the Mind” . In J. B. Copeland, C. Posy, O. Shagrir (editors) Computability: Gödel, Turing, Church, and beyond 1-35. Boston: MIT Press. DOI: 10.7551/mitpress/8009.003.0002
  12. Dawson, John. 1997. Logical Dilemmas. The Life and Work of Kurt Gödel. Wellesley: A K Peters. DOI: 10.1201/9780429294884
  13. Gödel, Kurt. 1951. “Some basic theorems on the foundations of mathematics and their implications” (Gibbs Lecture). In Gödel’s Collected Works (1986-2005) III, 304-323.
  14. Gödel, Kurt. 1931. “Über Formal Unentscheidbare Sätze der Principia Mathematica und Verwandter Systeme I.” Monatshefte für Mathematik und Physik 38: 173-198 (reprinted in Gödel’s Collected Works (1986-2005), Vol. I: 144-195 DOI: 10.1007/BF01700692
  15. Gold, E. Mark. 1967. “Language identification in the limit” Information and Control 10 (5): 447-474.
  16. Grattan-Guinness, Ivor. 1979. “In Memoriam Kurt Gödel: his 1931 correspondence with Zermelo on his incompletability theorem” . Historia Math. 6: 294-304.
  17. Grattan-Guinness, Ivor. 1979. “In Memoriam Kurt Gödel: his 1931 correspondence with Zermelo on his incompletability theorem” . Historia Math. 6: 294-304. DOI: 10.1016/J.HM.2015.09.002
  18. Grattan-Guinness, Ivor. 2011. “The Reception of Gödel’s 1931 Incompletability Theorems by Mathematicians, and Some Logicians, to the Early 1960s. ” In Kurt Gödel and the Foundations of Mathematics, edited by M. Baaz, 57-74. Cambridge: Cambridge University Press. DOI: 10.1017/CBO9780511974236
  19. Hilbert, David and Bernays, Paul. 1934. Grundlagen der Mathematik I. Berlin: Springer.
  20. Hilbert, David and Bernays, Paul. 1939. Grundlagen der Mathematik II. Berlin: Springer.
  21. Krajewski, Stanislaw. 2004. “Gödel on Tarski.” Annals of Pure and Applied Logic 127: 303- 323. DOI: 10.1016/j.apal.2003.11.019.
  22. Kelly, Kevin T. 2023. The Logic of Reliable Inquiry. Oxford: Oxford University Press. DOI: 10.1093/oso/9780195091953.003.0002
  23. Kugel, Peter. 1986. “Thinking may be more than computing.” Cognition 18: 128-149.
  24. Jeřábek, Emil. 2016. “Division by zero” Archive for Mathematical Logic 55: 997-1013. DOI: 10.1007/s00153-016-0508-5
  25. Jeroslow, Robert G. 1975. “Experimental logics and ∆ 2 theories.” Journal of Philosophical Logic 4 (3): 253-267 DOI: 10.1007/BF00261270
  26. Magari, Roberto. 1974. “Su certe teorie non enumerabili.” Annali di Matematica Pura ed Applicata 4, XCVIII: 119-152. DOI: 10.1007/BF02792706
  27. Lakatos, Imre. 1976. Proofs and Refutations. Cambridge: Cambridge University Press. DOI: 10.1017/CBO 9781139171472
  28. Matiyasevich, Yuri. 1970. "Enumerable sets are Diophantine." Doklady Akademii Nauk SSSR, 191: 79-282.
  29. Odifreddi, Piergiorgio. 1989. Classical Recursion Theory I. Amsterdam: North Holland.
  30. Osherson, Daniel. N. and Stob, Michael and Weinstein, Scott. 1986. Systems that learn: An introduction to learning theory for cognitive and computer scientists. Boston: MIT Press.
  31. Parsons, Charles. 1970. “On a Number Theoretic Choice Schema and its Relation to Induction.” in A. Kino, J. Myhill, R.E. Vesley (editors), Studies in Logic and the Foundations of Mathematics vol. 60. Amsterdam: Elsevier. DOI: 10.2307/2272677
  32. Peter, Rozsa. 1957. Rekursive Funktionen. Berlin, Boston: de Gruyter.
  33. Post, Emil. 1941. “Absolutely Unsolvable Problems and Relatively Undecidable Propositions: Account of an Anticipation.” In The Undecidable 1965, edited by M. Davis 340-433. New York: Raven Press.
  34. Post, Emil. 1944. “Recursively enumerable sets of positive integers and their decision problems”. Bulletin of the American Mathematical Society 50: 284-316.
  35. Post, Emil L. 1948. “Degrees of recursive unsolvability: preliminary report.” Bulletin of the American Math. Soc. 54: 641-642.
  36. Putnam, Hilary. 1965. “Trial and error predicates and the solution to a problem of Mostowski.” The Journal of Symbolic Logic 30 (1): 49-57. DOI: 10.2307/2270581
  37. Robinson, Raphael. 1950, “An Essentially Undecidable Axiom System”, in Proceedings of the International Congress of Mathematics, 1950: 729-730
  38. Rogers, Hartley. 1987. The Theory of Recursive Functions and Effective Computability. Boston: MIT Press.
  39. Smullyan, Raymond Merrill. 1961. Theory of Formal Systems. Princeton: Princeton University Press.
  40. Soare, Robert I. 1996. “Computability and Recursion.” The Bulletin of Symbolic Logic 2 (3): 284-321. DOI: 10.2307/420992
  41. Soare, Robert I. 2009. “Turing oracle machines, online computing, and three displacements in computability theory”. Annals of Pure and Applied Logic 160 (3): 368-399.
  42. Turing, Alan M. 1936. “On computable numbers, with application to the Entscheidungsprob- lem”, in Proceedings of London Math. Soc. 42: 230-265, 544-546.
  43. Turing, Alan M. 1947. “Lecture to the London Mathematical Society on 20 February 1947.” In The Essential Turing, edited by B. J. Copeland, 378-394. Oxford: Oxford U.P.
  44. Von Neumann, John. 1951. “The general and logical theory of automata”. In Cerebral mechanisms in behavior; the Hixon Symposium, edited by L. A. Jeffress, 1-41. Hoboken: Wiley.
  45. Wang, Hao. 1974. From Mathematics to Philosophy. New York: Routledge and Kegan.
  46. Zach, Richard. 1998. “Numbers and functions in Hilbert’s finitism”. Taiwanese Journal for Philosophy and History of Science 10: 33-60. DOI: 10.1017/S1755020320000015
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  • Publication Year: 2025
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Chapter Information

Chapter Title

Abstract views of incompleteness

Authors

Duccio Pianigiani

Language

English

DOI

10.36253/979-12-215-0778-2.05

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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