List of uncomputable numbers

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This is a comprehensive list of known uncomputable numbers. Uncomputable numbers are real numbers that cannot be computed by any algorithm in a finite amount of time. These numbers play a crucial role in the theory of computation and mathematical logic.

Introduction[edit]

Uncomputable numbers are real numbers that cannot be computed by any algorithm in a finite amount of time. They are closely related to the concept of undecidability in computer science and mathematical logic. This list aims to provide a comprehensive overview of known uncomputable numbers and their significance.

List of uncomputable numbers[edit]

  1. Chaitin's constant (Ω): The halting probability of a universal Turing machine.
  2. Busy Beaver numbers: Related to the Busy Beaver function, which grows faster than any computable function.
  3. Kolmogorov complexity of strings: The length of the shortest program that produces a given string.
  4. Rayo's number: Defined as the largest number expressible in first-order set theory with a googol symbols.
  5. TREE(3): A number derived from Kruskal's tree theorem in graph theory.
  6. SCG(13): The 13th number in the Sussex-Cohen sequence of "large" numbers.
  7. Radó's Sigma function: A variation of the Busy Beaver function.
  8. Gödel's incompleteness constant: Related to the encoding of Gödel's incompleteness theorems.
  9. ZFC provability probability: The probability that a random statement in ZFC set theory is provable.
  10. Degrees of unsolvability: Numbers representing the computational power of certain problems.
  11. Scott's constant: Related to the λ-calculus and computability theory.
  12. Paris-Harrington numbers: Derived from the Paris-Harrington theorem in Ramsey theory.
  13. Friedman's TREE sequence: An extension of TREE(3) to higher orders.
  14. Beklemishev's worm numbers: Related to proof theory and ordinal analysis.
  15. Goodstein sequence limits: The limits of certain rapidly growing sequences.
  16. Doodle numbers: Related to certain combinatorial problems in graph theory.
  17. Fast-growing hierarchy limits: Limits of functions in the fast-growing hierarchy.
  18. Ackermann numbers: Numbers derived from the Ackermann function.
  19. Takeuti-Feferman-Buchholz ordinals: Ordinals related to proof-theoretic strength.
  20. Kruskal's constant: Related to Kruskal's tree theorem and Harvey Friedman's work.
  21. Saibian's Hyperfactorial Array Notation (HAN) limits: Limits of certain array notation systems.
  22. Bird's array notation limits: Limits of Bird's array notation system.
  23. Loader's number: Related to the halting problem for simply typed λ-calculus.
  24. Skewes' number: The smallest solution to a certain inequality involving prime numbers.
  25. Graham's number: While not strictly uncomputable, it's so large that its exact value cannot be known.
  26. Mega: A number defined using Knuth's up-arrow notation, related to large number notations.
  27. Moser's number: A large number arising from a problem in spherical geometry.
  28. SSCG(3): The third number in the Super Sussex-Cohen sequence.
  29. Friedman's finite promise games numbers: Arising from certain two-player games.
  30. Hoogerbrug's number: A number defined using nested Conway chained arrow notation.

Significance and implications[edit]

The existence of uncomputable numbers has profound implications for mathematics, computer science, and philosophy. They demonstrate fundamental limits to what can be calculated algorithmically and highlight the differences between human mathematical reasoning and mechanical computation. Uncomputable numbers play crucial roles in various areas of mathematical logic, including proof theory, set theory, and the foundations of mathematics.

See also[edit]

References[edit]

  1. Chaitin, G. J. (1975). "A theory of program size formally identical to information theory". Journal of the ACM, 22(3), 329-340.
  2. Rado, T. (1962). "On non-computable functions". Bell System Technical Journal, 41(3), 877-884.
  3. Kolmogorov, A. N. (1965). "Three approaches to the quantitative definition of information". Problems of Information Transmission, 1(1), 1-7.
  4. Friedman, H. (1982). "Boolean relation theory and incompleteness". Annals of Mathematics Logic, 23(1), 1-14.
  5. Beklemishev, L. D. (2003). "Proof-theoretic analysis by iterated reflection". Archive for Mathematical Logic, 42(6), 515-552.