1998
DOI: 10.1023/a:1004305315546
|View full text |Cite
|
Sign up to set email alerts
|

On Interpreting Chaitin's Incompleteness Theorem

Abstract: The aim of this paper is to comprehensively question the validity of the standard way of interpreting Chaitin's famous incompleteness theorem, which says that for every formalized theory of arithmetic there is a finite constant c such that the theory in question cannot prove any particular number to have Kolmogorov complexity larger than c. The received interpretation of theorem claims that the limiting constant is determined by the complexity of the theory itself, which is assumed to be good measure of the st… Show more

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...
2
1
1

Citation Types

0
17
0
5

Year Published

2008
2008
2021
2021

Publication Types

Select...
5
2

Relationship

0
7

Authors

Journals

citations
Cited by 30 publications
(29 citation statements)
references
References 11 publications
0
17
0
5
Order By: Relevance
“…An additional nice property of using physical stochastic models, e.g., statistical mechanics, stochastic dynamical systems, quantum computing models, instead of abstract machine or computation models is that we can refute a wellknown objection to algorithmic information by Raatikainen [11], which depends on unnatural enumerations of recursive functions, essentially constructing reference machines with a lot of useless information. Such superfluous reference machines would incur a physical cost in physical message complexity, and therefore they would not be picked by our definition, which is exactly why you cannot shuffle program indices as you like, because such permutations require additional information to encode.…”
Section: Refuting the Platonist Objection To Algorithmic Informationmentioning
confidence: 99%
“…An additional nice property of using physical stochastic models, e.g., statistical mechanics, stochastic dynamical systems, quantum computing models, instead of abstract machine or computation models is that we can refute a wellknown objection to algorithmic information by Raatikainen [11], which depends on unnatural enumerations of recursive functions, essentially constructing reference machines with a lot of useless information. Such superfluous reference machines would incur a physical cost in physical message complexity, and therefore they would not be picked by our definition, which is exactly why you cannot shuffle program indices as you like, because such permutations require additional information to encode.…”
Section: Refuting the Platonist Objection To Algorithmic Informationmentioning
confidence: 99%
“…The previous statement, and Chaitin's assertion that the Kolmogorov complexity of T somehow measures the power of T as a theory, has been much criticized in van Lambalgen (1989), Fallis (1996) and Raatikainen (1998). The previous statement, and Chaitin's assertion that the Kolmogorov complexity of T somehow measures the power of T as a theory, has been much criticized in van Lambalgen (1989), Fallis (1996) and Raatikainen (1998).…”
Section: Definition 324 (Chaitin 1974) Kmentioning
confidence: 99%
“…Raatikainen [5] focused on the choice of universal Turing machine as a factor which influences the value of c T and he proved that for any recursively axiomatizable consistent theory T there exists a universal Turing machine such that c T is zero; thus c S = c T is always possible for two theories S and T.…”
Section: Introductionmentioning
confidence: 99%
“…Another characteristic constant r T was focused on by Raatikainen [5] in connection with c T . We show that the difference between c T and r T can be made arbitrarily large by chainging the universal Turing machine, but, on the other hand, that r S < r T for some universal Turing machine if and only if c S < c T for some (possibly different) universal Turing machine.…”
Section: Introductionmentioning
confidence: 99%