## Information-theoretic Limitations of Formal Systems (1974)

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Venue: | JOURNAL OF THE ACM |

Citations: | 47 - 8 self |

### BibTeX

@ARTICLE{Chaitin74information-theoreticlimitations,

author = {Gregory J. Chaitin},

title = {Information-theoretic Limitations of Formal Systems},

journal = {JOURNAL OF THE ACM},

year = {1974},

volume = {21},

pages = {403--424}

}

### Years of Citing Articles

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### Abstract

An attempt is made to apply information-theoretic computational complexity to metamathematics. The paper studies the number of bits of instructions that must be a given to a computer for it to perform finite and infinite tasks, and also the amount of time that it takes the computer to perform these tasks. This is applied to measuring the difficulty of proving a given set of theorems, in terms of the number of bits of axioms that are assumed, and the size of the proofs needed to deduce the theorems from the axioms.

### Citations

838 | Theory of Recursive Functions and Effective Computability - ROGERS - 1967 |

458 | L.: Computation: Finite and Infinite Machines - Minsky - 1967 |

330 | A theory of program size formally identical to information theory - Chaitin - 1975 |

306 | Introduction to Mathematical Logic - Mendelson - 1997 |

216 | Mathematical Logic - Shoenfield - 1967 |

185 | The Complexity of Finite Objects and the Development of the Concepts of Inforrna- p. 73. tion and Randomness by Means of the Theory of Algo- rithms - Zvonkin, Levin - 1970 |

152 | Zufälligkeit und Wahrscheinlichkeit. Eine algorithmische Begründung der Wahrscheinlichkeitstheorie - Schnorr |

118 | Mathematical logic as based on the theory of types - Russell - 1908 |

117 |
Recursively enumerable sets of positive integers and their decision problems
- Post
- 1944
(Show Context)
Citation Context ...esults to formal systems. This section explains how this is done. The abstract definition used by Post that a formal system is an r.e. set of propositions is close to the viewpoint of this paper (see =-=[31]-=-). 6 6 For standard definitions of formal systems, see, for example, [32--34] and [10, p. 117]. 12 G. J. Chaitin However, we are not quite this unconcerned with the internal details of formal systems.... |

115 | Computability and Unsolvability - Davis |

94 |
Set Theory and the Continuum Hypothesis
- Cohen
- 1966
(Show Context)
Citation Context ... Theorem 4.1 of this paper. 4 [13--16] are related in approach to this paper. [13, 15, and 16] are concerned with measuring the size of proofs and the effect of varying the axioms upon their size. In =-=[14] Cohen &qu-=-ot;measures the strength of a [formal] system by the ordinals which can be handled in the system." 5 The analysis that follows of the possible significance of the results of this paper has been i... |

91 | Logical Basis for Information Theory and Probability Theory - Kolmogorov - 1968 |

80 |
Hilbert’s tenth problem is unsolvable
- Davis
- 1973
(Show Context)
Citation Context ...er of bits of axioms that are assumed and what can be deduced. This example is based on the work of M. Davis, Ju. V. Matisjasevic, H. Putnam, and J. Robinson that settled Hilbert's tenth problem (cf. =-=[30]-=-). There is a polynomial P in k+2 variables with integer coefficients that has the following property. Consider the infinite string whose ith bit is 1 or 0 depending on whether or not the set S i = fn... |

64 | Mathematical logic - Kleene - 1967 |

44 | A variant of the kolmogorov concept of complexity - Loveland - 1969 |

37 |
What is Cantor’s continuum problem
- Gödel
- 1947
(Show Context)
Citation Context ...or deciding, given any p, whether or not the string 1 is in U(p). Added in Proof The following additional references have come to our attention. Part of Gödel’s analysis of Cantor’s continuum pro=-=blem [39] i-=-s highly relevant to the philosophical considerations of Section 1. Cf. especially [39, pp. 265, 272]. Schwartz [40, pp. 26–28] first reformulates our Theorem 4.1 using the hypothesis that the forma... |

35 | Information-Theoretic Computational Complexity
- Chaitin
- 1974
(Show Context)
Citation Context ...ere granted by permission of the Association for Computing Machinery. An early version of this paper was presented at the Courant Institute Computational Complexity Symposium, New York, October 1971. =-=[28]-=- includes a nontechnical exposition of some results of this paper. [1] and [2] announce related results. Author's address: Rivadavia 3580, Dpto. 10A, Buenos Aires, Argentina. Information-Theoretic Lim... |

22 | Computational complexity and probability construction - Willis |

21 |
Philosophy of Mathematics and Natural Science
- Weyl
- 1949
(Show Context)
Citation Context ...In [3] and [4] von Neumann analyzes the effect of Godel's theorem upon mathematicians. Weyl's reaction to Godel's theorem is quoted by Bell [5]. The original source is [6]. See also Weyl's discussion =-=[7]-=- of Godel's views regarding his incompleteness theorem. 3 For nontechnical expositions of Godel's incompleteness theorem, see [8, 9, 10, Sec. 1, pp. xv-xviii, 11, and 12]. [28] contains a nontechnical... |

20 | On the Difficulty of Computations - Chaitin - 1970 |

20 |
Computer Studies of Turing Machine Problems
- Lin, Rado
- 1965
(Show Context)
Citation Context ...upper bound on a(n + c), one faces the same drastic alternatives. Lin and Rado, in trying to determine particular values of \Sigma(n) and SH(n), have, in fact, essentially been trying to do this (see =-=[36]-=-). In their paper they explain the difficulties they encountered and overcame for n = 3, and expect to be insurmountable for greater values of n. 7.2 Now we begin the formal exposition, which is couch... |

15 | Abbreviating proofs by adding new axioms - Ehrenfeucht, Mycielski - 1971 |

15 | On the notion of randomness - Martin-Löf - 1968 |

10 | On the lengths of proofs - Gödel - 1936 |

9 | O n Programming: An Interim Report on - Schwartz |

9 | Heuristic reasoning in the theory of numbers - Pólya - 1959 |

8 |
A Profile of Mathematical Logic
- DeLong
- 1970
(Show Context)
Citation Context ...e will be an enormously great number of bits of axioms (Theorem 7.6(c)). Another theme of this paper arises from the following metamathematical considerations, which are well known (see, for example, =-=[29]-=-). In a formal system without a decision method, it is impossible to bound the size of a proof of a theorem by a recursive function of the number of characters in the statement of the theorem. For if ... |

8 | Mathematical logic as based on the theory of types, From Frege to Godel: A source book in mathematical logic - Russell - 1967 |

7 | Remarks on Bertrand Russell's theory of knowledge. Libr. of Living Philos - Einstein - 1946 |

5 |
The mathematician
- Neumann
- 1947
(Show Context)
Citation Context ...oach to metamathematics; 4 a great deal of work remains to be done to clarify these matters. Nevertheless, we would like to sketch here the conclusions which we have tentatively drawn. 5 2 In [3] and =-=[4]-=- von Neumann analyzes the effect of Godel's theorem upon mathematicians. Weyl's reaction to Godel's theorem is quoted by Bell [5]. The original source is [6]. See also Weyl's discussion [7] of Godel's... |

4 | Solvable and Unsolvable Problems - Turing - 1954 |

4 | Speed-up theorems and incompleteness theorems - Arbib - 1966 |

4 | Theories of Probability-An Examination of Foundations - Fine - 1973 |

4 | On minimal-program complexity measures - Loveland - 1969 |

4 |
What is Cantor's continuum problem
- odel
- 1964
(Show Context)
Citation Context ...or deciding, given any p, whether or not the string 1 is in U(p). Added in Proof The following additional references have come to our attention. Part of Godel's analysis of Cantor's continuum problem =-=[39]-=- is highly relevant to the philosophical considerations of Section 1. Cf. especially [39, pp. 265, 272]. Schwartz [40, pp. 26--28] first reformulates our Theorem 4.1 using the hypothesis that the form... |

3 | Mathematical sieves - Hawkins - 1958 |

2 |
Method in the Physical Sciences
- Neumann
- 1955
(Show Context)
Citation Context ... an approach to metamathematics; 4 a great deal of work remains to be done to clarify these matters. Nevertheless, we would like to sketch here the conclusions which we have tentatively drawn. 5 2 In =-=[3]-=- and [4] von Neumann analyzes the effect of Godel's theorem upon mathematicians. Weyl's reaction to Godel's theorem is quoted by Bell [5]. The original source is [6]. See also Weyl's discussion [7] of... |

1 |
Information-theoretic aspects of Post’s construction of a simple set. On the difficulty of generating all binary strings of complexity less than n
- Chaitin
- 1972
(Show Context)
Citation Context ...n early version of this paper was presented at the Courant Institute Computational Complexity Symposium, New York, October 1971. [28] includes a nontechnical exposition of some results of this paper. =-=[1]-=- and [2] announce related results. Author's address: Rivadavia 3580, Dpto. 10A, Buenos Aires, Argentina. Information-Theoretic Limitations of Formal Systems 3 to be able to obtain a set of theorems? H... |

1 |
On the greatest natural number of definitional or information complexity ≤ n. There are few minimal descriptions. (Abstracts.) Recursive Function Theory
- Chaitin
- 1973
(Show Context)
Citation Context ...version of this paper was presented at the Courant Institute Computational Complexity Symposium, New York, October 1971. [28] includes a nontechnical exposition of some results of this paper. [1] and =-=[2]-=- announce related results. Author's address: Rivadavia 3580, Dpto. 10A, Buenos Aires, Argentina. Information-Theoretic Limitations of Formal Systems 3 to be able to obtain a set of theorems? How long ... |

1 |
Mathematics: Queen and Servant of Science
- Bell
- 1951
(Show Context)
Citation Context ...h here the conclusions which we have tentatively drawn. 5 2 In [3] and [4] von Neumann analyzes the effect of Godel's theorem upon mathematicians. Weyl's reaction to Godel's theorem is quoted by Bell =-=[5]-=-. The original source is [6]. See also Weyl's discussion [7] of Godel's views regarding his incompleteness theorem. 3 For nontechnical expositions of Godel's incompleteness theorem, see [8, 9, 10, Sec... |

1 | On the length of proofs - odel, K - 1965 |

1 | Heuristic reasoning in the theory of - olya, G - 1959 |

1 | Algorithms and randomness - of, P - 1969 |

1 | A short survey of computational complexity theory - Schwartz - 1972 |