Contained in:
Book Chapter

Random sequences, incompleteness and information

  • Duccio Pianigiani

Another mathematically significant development in the research around the phenomenon of incompleteness on which we consider it important to focus attention, is that which has led to highlighting the link between this concept and that of randomness and information. Grigory Chaitin, starting in the 1970s, reformulated the incompleteness theorems within the framework of algorithmic theory of information, presenting his results as a “dramatic extension” of the phenomenon already highlighted by Gödel. He claims that high complexity is the alleged reason of the unprovability of infinitely many true sentences: a true statement that can be expressed in the language of a theory is unprovable because its information content is greater than that of the axioms of that theory itself; or in other words, in a formal system no number can be proved random unless its complexity is less than that of the formal system itself. At the heart of Chaitin’s work, therefore, is the concept of the complexity of an object and the measure of the difficulty of describing it. The mathematical concept of randomness is an attempt to give an idealized model of randomness, as the recursive functions do in the case of computability.

  • Keywords:
  • Kolmogorov-Solomonoff complexity,
  • Martin Loef test,
  • algorithmic randomness,
+ Show More

Duccio Pianigiani

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

  1. Bailly, Francis and Longo, Giuseppe. 2007. “Randomness and determinism in the interplay between the continuum and the discrete” . Mathematical Structures in Computer Science 17: 289-305. DOI: 10.1017/S0960129507006007
  2. Bailly, Francis and Longo, Giuseppe. 2008. “Phenomenology of Incompleteness: from Formal Deductions to Mathematics and Physics” . In R. Lupacchini (editor) Deduction, Computa- tion, Experiment Berlin: Springer.
  3. Barmpalias, George and Lewis-Pye, Andrew. 2018. “Optimal redundancy in computations from random oracles” . Journal of Computer and System Sciences 92: 1-8. DOI: 10.1016/j.jcss.2017.06.009
  4. Barzdins, Janis. 1968. “Complexity of programs to determine whether natural numbers not greater than n belong to a recursively enumerable set”. Soviet Math. Dokl. 9: 1251-54.
  5. Borel, Émile. 1909. “Les probabilités dénombrables et leurs applications arithmétiques” . Rendiconti del Circolo Matematico di Palermo 27: 247-271.
  6. Calude, Cristian ed. 2007. Randomness and Complexity, from Leibniz to Chaitin. Singapore: World Scientific.
  7. Calude, Claude. 1994. “Borel normality and algorithmic randomness” . In G. Rozenberg, A. Salomaa (editors) Developments in Language Theory 113-129. Singapore: World Scientific.
  8. Calude, Cristian S. and Jürgensen, Helmut. 2005. “Is complexity a source of incompleteness?”. Advances in Applied Mathematics 35 (1): 1-15. DOI: 10.1016/j.aam.2004.10.003
  9. Calude, Christian S. and Stay, Michael. 2007. “From Heisenberg to Gödel via Chaitin”. International Journal of Theoretical Physics 46 (8): 1053-1065. DOI: 10.1007/s10773-006-9296-8
  10. Chaitin, Gregory J. 1974. “Information-theoretic limitations of formal systems” Journal of the ACM 21 (3): 403-424. DOI: 10.1145/321832.321839
  11. Chaitin, Gregory. 1977. “Algorithmic Information Theory”. IBM Journal of Research and Development 21 (4): 50-35
  12. Chaitin, Gregory. 1982. “Gödel’s theorem and information” . International Journal of Theoret. Physics 22: 941-954. DOI: 10.1007/BF02084159
  13. Chaitin, Gregory. 1990. Information, Randomness and Incompleteness, Papers on Algorithmic Information Theory (second edition). Singapore: World Scientific.
  14. Chaitin, Gregory. 2003. Conversations with a Mathematician. Math, Art, Science and the Limits of Reason. Berlin: Springer.
  15. Chaitin, Gregory. 2007. Thinking of Gödel and Turing. Singapore: World Scientific. DOI: 10.1142/6536
  16. Church, Alonzo. 1940. “On the concept of a random sequence” . Bulletin of the American Mathematical Society 46: 130-135. DOI: 10.2307/2266178
  17. Da Costa, Newton and Doria, Francisco A. 1991. “Undecidability and Incompleteness in Classical Mechanics” . International Journal of Theoretical Physics 30: 1041-1073. DOI: 10.1007/BF00671484
  18. Demuth, Osvald. 1982. “Some classes of arithmetical real numbers ” (in Russian). Commen- tationes Mathematicae Universitatis Carolinae. 23 (3): 453-465.
  19. Downey, Rod and Hirschfeldt, Denis. 2010. Algorithmic Randomness and Complexity. Berlin: Springer.
  20. Downey, Rod and Hirschfeldt, Denis. 2019. “Computability and Randomness” Notices of the American Mathematical Society. 66: 1001-1012. DOI: 10.1090/noti1905
  21. Endrullis, Jörg and Klop, Jan and Overbeek, Roy. 2021. “Star Games and Hydras.”Logical Methods in Computer Science 17 (2):1-32. DOI: 10.1016/S0304-3975(02)00040-3
  22. Ferbus-Zanda, Marie and Grigorieff, Serge. 2008. “Is Randomness Native to Computer Science?” Bulletin of The European Association for Theoretical Computer Science 74: 78-118. DOI: 10.1142/9789812562494_0046
  23. Ferbus-Zanda, Marie and Grigorieff, Serge. 2014.“Kolmogorov Complexity in Perspective Part I: Information Theory and Randomness”. In Constructivity and Computability in Historical and Philosophical Perspective. Logic, Epistemology, and the Unity of Science 34, edited by J. Dubucs and M. Bourdeau 57-94. Dordrecht: Springer Netherland.
  24. Franzen, Torkel. 2005. Gödel’s theorem: an incomplete guide to its use and abuse (1st ed.). Wellesley: A. K. Peters. DOI: 10.1201/9780429295225
  25. Gács, Peter. 1986. “Every sequence is reducible to a random one” . Information and Control 70 (2–3): 186-192. DOI: 10.1016/S0019-9958(86)80004-3
  26. Hammer, Daniel and Romashchenko, Andrei and Shen, Alexander and Vereshchagin, Niko- lay. 2000. “Inequalities for Shannon entropies and Kolmogorov complexities”. Journal of Computer and System Sciences 60 (2): 442-464. DOI: 10.1109/CCC.1997.612296
  27. Jech, Thomas. 1978. Set theory. Cambridge M.: Academic Press. DOI: 10.2307/2273242
  28. Lafitte, Grégory. 2002. “On Randomness and Infinity”. In Foundations of Information Tech- nology in the Era of Network and Mobile Computing. IFIP-The International Federation for Information Processing, edited by R. Baeza-Yates, U. Montanari, M. Santoro, 267-279. Boston: Springer US DOI: 10.1007/978-0-387-35608-2_23
  29. Levin, Leonid. 1973. “Universal search problems” . Problems of Information Transmission 9 (3): 115-116.
  30. Kamke, Erich. 1932. “Über neuere Begriindungen der Wahrscheinlichkeitsrechnung” . Jahres- bericht der Deutschen Mathematiker- Vereinigung 42: 121-149.
  31. Katseff, Howard. 1978. “Complexity dips in random infinite binary sequences”. Information and Control 38: 258-263. DOI: 10.1016/S0019-9958(78)90062-1
  32. Kolmogorov, Andrej N. 1965. “Three Approaches to the quantitive definition of Informa- tion”. Problems of Information Transmission 1: 1-17. DOI: 10.1080/00207166808803030
  33. Kreisel, Georg. 1969. “Two notes on the foundations of set theory. ”, Dialectica 23: pp. 93-114.
  34. Kučera, Antonin. 1985. “Measure, Π 01 -classes and complete extensions of PA.” In Recursion Theory Week. Lecture Notes in Mathematics 1141, edited by H.D. Ebbinghaus, G.H. Müller and G.E. Sacks 245-259. Berlin: Springer.
  35. Martin-Löf, Per. 1966. “The definition of random sequences” . Information and Control 9: 602-619. DOI: 10.1016/S0019-9958(66)90261-9
  36. Moore, Cristopher. 1990. “Unpredictability and undecidability in dynamical systems.” Physical review letters 64 (2): 2354-2357. DOI: 10.1103/PhysRevLett.64.2354
  37. Nies, André. 2009. Computability and Randomness. Oxford: Oxford University Press. DOI: 10.2178/bsl/1264433798
  38. Nies, André and Scholz, Volkher B. 2017. “Martin-Löf random quantum states” . Journal of Mathematical Physics 60 (9): 1-11. DOI: 10.1063/1.5094660
  39. Peres, Asher. 1985. “Einstein, Gödel, Bohr” Foundations of Physics 15: 201-205. DOI: 10.1007/BF00735292
  40. Peres, Asher and Zurek, Wojciech H. 1982. “Is quantum theory universally valid?”. American Journal of Physics 50 (9): 807-810. DOI: 10.1119/1.13086
  41. Raatikainen, Panu. 1998. “On Interpreting Chaitin’s Incompleteness Theorem”. Journal of Philosophical Logic 27 (6): 569-586. DOI: 10.1023/A:1004305315546
  42. Raatikainen, Panu. 2000. “Algorithmic information theory and undecidability” . Synthese 123 (2): 217-225. DOI: 10.1023/A:1005298819345
  43. Schnorr, Claus-Peter. 1971. “A unified approach to the definition of random sequences.”. Mathematical Systems Theory 5: 246-258. DOI: 10.1007/BF01694181
  44. Schnorr, Claus Peter. 1973. “Process complexity and effective random tests”. Journal of Computer and System Sciences 7 (4,) 376-388. DOI: 10.1016/S0022-0000(73)80030-3
  45. Shannon, Claude. 1984. “A mathematical theory of communication”. Bell System Technical Journal 27: 379-423 and 623-656. DOI: 10.1002/j.1538-7305.1948.tb01338.x
  46. Solovay, Robert. 1970. “A Model of Set Theory in Which Every Set of Reals is Lebesgue Measurable”. Annals of Mathematics. Second Series: 1-56. DOI: 10.2307/1970696
  47. Solovay, Robert. 2000. “A Version of Ω for which ZFC can not Predict a Single Bit”. In Finite Versus Infinite. Contributions to an Eternal Dilemma, edited by C. Calude and G. Paŭn 323-334. Berlin: Springer. DOI: 10.1007/s10773-006-9296-8
  48. Solomonoff, Ray. 1964. “A formal theory of inductive inference. I”. Information and Control 7: 1-22. DOI: 10.1016/S0019-9958(64)90223-2
  49. Solomonoff, Ray. 1964. “A formal theory of inductive inference. II”. Information and Control 7: 224-254. DOI: 10.1016/S0019-9958(64)90131-7
  50. Stephan, Frank. 2006. “Martin-Löf random and PA-complete sets” . In Logic Colloquium ’02. Lecture Notes in Logic, edited by Z. Chatzidakis, P. Koepke and W. Pohlers W. 342-348. Cambridge: Cambridge University Press. DOI: 10.1017/9781316755723.016
  51. Towsner Henry. 2020. “Algorithmic randomness in ergodic theory”. In Algorithmic Random- ness: Progress and Prospects. Lecture Notes in Logic, edited by J.N.Y. Franklin and C.P. Porter 40-57. Cambridge: Cambridge University Press. DOI: 10.1017/9781108781718.003
  52. Van Lambalgen, Michiel. I 1987. Random Sequences. Doctoral thesis, University of Amsterdam.
  53. Van Lambalgen, Michiel. II 1987. “Algorithmic information theory” The Journal of Symbolic Logic 54 (4): 1389-1400. DOI: 10.2307/2274821
  54. Van Lambalgen, Michiel. 1987. “Von Mises’ Definition of Random Sequences Reconsidered.” The Journal of Symbolic Logic 52 (3): 725-755 DOI: 10.2307/2274360
  55. Van Lambalgen, Michiel. 1990. “The axiomatization of randomness” The Journal of Symbolic Logic 55 (3): 1143-1167. DOI: 10.2307/2274480
  56. Van Lambalgen, Michiel. 1992. “Independence, randomness and the axiom of Choice” . The Journal of Symbolic Logic 57 (4): 1274-1304. DOI: 10.2307/2275368
  57. V’yugin, Vladimir. 2022. “Ergodic theorems for algorithmically random points”. 10.48550/arXiv.2202.13465. DOI: 10.48550/arXiv.2202.13465
  58. Zaffora Blando, Francesca. 2024. “From Wald to Schnorr: Von Mises’ definition of randomness in the aftermath of Ville’s Theorem.”. Studies in History and Philosophy of Science 106: 196-207. DOI: 10.1071/HR18014
PDF
  • Publication Year: 2025
  • Pages: 205-224

XML
  • Publication Year: 2025

Chapter Information

Chapter Title

Random sequences, incompleteness and information

Authors

Duccio Pianigiani

Language

English

DOI

10.36253/979-12-215-0778-2.13

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

0

Fulltext
downloads

0

Views

Export Citation

1,428

Open Access Books

in the Catalogue

2,808

Book Chapters

4,874,303

Fulltext
downloads

5,185

Authors

from 1115 Research Institutions

of 66 Nations

72

scientific boards

from 401 Research Institutions

of 44 Nations

1,314

Referees

from 407 Research Institutions

of 39 Nations