Add new comment
-
Want facts and want them fast? Our Maths in a minute series explores key mathematical concepts in just a few words.
A game you're almost certain to lose...
What are the challenges of communicating from the frontiers of mathematical research, and why should we be doing it?
Celebrate Pi Day with the stars of our podcast, Maths on the move!
Maths meets politics as early career mathematicians present their work at the Houses of Parliament.
Celebrate this year's International Women's Day with some of the articles and podcasts we have produced with women mathematicians over the last year!
At each point we have considered only a finite number of strings. Then it is always possible to add many times that many to tilt the probability back and forth. Note "halts for the first few inputs, loops on the next inputs for a lot more inputs".
What is the probability that the program that I describe will halt? There is none. "Probability of Halting" is as ill-defined as almost everything Chaitin says. His first version of omega was >1 (depending on how the Turing Machine is encoded!) which took him 20 years to realize and add a kludge rule about programs being inside of other programs. Now how does he know what THAT will produce?
His Invariance Theorem (that is said to be the foundation of his theory) is false. There is no bound between the lengths of the shortest program to perform a given function in two different programming languages (to justify using "length of the shortest program") because one language could require each character to be repeated 2 or more times due to its use over unreliable communications lines, and so the length can differ by any factor or absolute difference. "Length of the shortest program" is simple-minded nonsense - just like his "This is unprovable." use, the extent of his understanding of Godel - which is only the weaker Godel theorem based on Soundness and still weaker than Rosser's based on consistency - which is the maximum possible because in an inconsistent system every sentence is provable so there is no sentence that is undecidable.