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Omega is an amazing number. We know it has a digit in its binary expansion that is 0 or 1, yet it is impossible to formulate why. It is impossible to come up with a train of thought that explains it. There is no explanation that can be written down on a piece of paper.

So we are left with the weird conclusion:

1. It has the value it has for no reason.

2. It has the value it has for a reason that is impossible to formulate.

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Nah, Omega is just defined as a number which contains countably infinite bits of algorithmic information. Since algorithmic information is equivalent to thermodynamic information (and likewise relates to quantum information if you head in that direction...), the whole "you can't calculate Omega" thing is really just a way of saying, "You can't have an infinite-precision physical measurement encoded into a finite physical system."

Chaitin makes far too much metaphysics of his work.

I don't really understand the insight with Chaitin's number. It's unknowable, but so are a lot of things. The halting problem already says that there are programs which we can't know if they will halt or not. And before that, Godel showed there are simple logical sentences which are true and unprovable.

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