Another question: What gives us confidence that omega isn't zero, to an arbitrarily high precision? The fraction of programs that halts may be negligibly small...and then the number zero would contain no information at all about mathematics.
There's a lot I don't understand about this. You can know for sure that a program is going to halt, but you never know that it WON'T halt at some point in the future. So you can't just start with small programs and go one-by-one to see if they halt or not.
Instead, you run the first program one step, then you run the first two programs 2 steps, then the first three programs 3 steps, etc.
But you might run the first program 10 steps, the second program 20 steps, the third program 30 steps, etc. How do you know that this procedure will eventually converge to the same omega as the other procedure?
For that matter, how do you know that you won't run into a very different pattern when programs have length 10 billion characters, and the algorithm that looked like it was converging for the first 9 billion characters now starts to move toward a very different number?
Another question: What gives us confidence that omega isn't zero, to an arbitrarily high precision? The fraction of programs that halts may be negligibly small...and then the number zero would contain no information at all about mathematics.
https://www.worldscientific.com/doi/abs/10.1142/S0218127407018130
There's a lot I don't understand about this. You can know for sure that a program is going to halt, but you never know that it WON'T halt at some point in the future. So you can't just start with small programs and go one-by-one to see if they halt or not.
Instead, you run the first program one step, then you run the first two programs 2 steps, then the first three programs 3 steps, etc.
But you might run the first program 10 steps, the second program 20 steps, the third program 30 steps, etc. How do you know that this procedure will eventually converge to the same omega as the other procedure?
For that matter, how do you know that you won't run into a very different pattern when programs have length 10 billion characters, and the algorithm that looked like it was converging for the first 9 billion characters now starts to move toward a very different number?