Add new comment
-
Want facts and want them fast? Our Maths in a minute series explores key mathematical concepts in just a few words.
Weird and wonderful things can happen when you set a ball in motion on a billiard table — and the theory of mathematical billiards has recently seen a breakthrough.
Was vaccinating vulnerable people first a good choice? Hindsight allows us to assess this question.
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!
I agree that Chaitin has not provided a proof that there is no TOE. Furthermore, almost none of what he says is well-defined. Omega itself is not a number, but rather a function of an arbitrary universal Turing Machine.
Chaitin says that the Godel Sentence is true but for no reason since Mathematics is actually random, so there is no proof of it. But the Godel Sentence is true because of how it is constructed, and we can in fact prove it true - Godel proves it true or else his article would be worthless, a theorem without a proof - we simply can't prove it using Godel's formal system. Godel himself says that what his formal system cannot prove can be proven using metamathematics.
Chaitin says that he has a better proof of incompleteness than Godel, but Rosser already did that by proving a stronger theorem. Godel's proof requires w-consistency, but Rosser's proof works with any consistent system, which includes all w-consistent systems and also others. It is a stronger result. So it makes no sense to offer more proofs of a weaker theorem. Rosser's theorem is stronger.
Chaitin says that Omega is the chance that a random Turing Machine will halt. Whatever way he defines a number, it cannot be the probability that a random Turing Machine will halt because there is no such probability. The notion of that being a probability is not well-defined. We can easily construct a Turing Machine (program) that halts for the first few inputs, loops on the next inputs for a lot more inputs, halts on the next inputs for even more inputs etc. so the chance that it halts fluctuates between 1/3 and 2/3, depending on how many inputs you consider. It diverges rather than converges.
Chaitin says that he learned Godel's proof as a child, but he has never discussed the actual proof based on w-consistency, or even mentioned Rosser's proof. Furthermore, even when he talks about the far simpler Turing proof of the Unsolvability of the Halting Problem, he gets it wrong. He says that a program that would tell if another program halts could be run on itself. But that program has an input, while the input of that program is a single program with no input. What Turing actually defined was a program that halts if its input does not halt on itself, and loops if its input does halt on itself. The input is a single program because that program's input is itself.