Well, as I understand it Chaitin numbers are computable with a Zeno machine or an oracle machine with a halting oracle as they are Delta_20 in the arithmetic hierarchy. This still seems to follow my previous line of thinking; that reasoning about incomputable mathematics is itself computable (of course), but (more interestingly) can be framed as reasoning with counterfactuals about physical law (what if I could break the speed of light, in this case).
I would also note that the reals would be countable if it weren’t for incomputable numbers. Any number that can be approximated to an arbitrary degree by an algorithm can of course be represented by that algorithm, and thus the set of all such numbers is equivalent to a subset of Turing machines, which are countable.
All this considered, I’m not really sure how Chaitin’s constant (at least, in particular, beyond the fact that it is incomputable) bears on the issue.
Well, as I understand it Chaitin numbers are computable with a Zeno machine or an oracle machine with a halting oracle as they are Delta_20 in the arithmetic hierarchy. This still seems to follow my previous line of thinking; that reasoning about incomputable mathematics is itself computable (of course), but (more interestingly) can be framed as reasoning with counterfactuals about physical law (what if I could break the speed of light, in this case).
I would also note that the reals would be countable if it weren’t for incomputable numbers. Any number that can be approximated to an arbitrary degree by an algorithm can of course be represented by that algorithm, and thus the set of all such numbers is equivalent to a subset of Turing machines, which are countable.
All this considered, I’m not really sure how Chaitin’s constant (at least, in particular, beyond the fact that it is incomputable) bears on the issue.
I think I was confused about what you were confused about.