So, as I understand it, computable numbers are countable since you can order the set of Turing machines that compute a number based on their Gödel number and thus count the numbers they compute. This means there are "as many" numbers that cannot be computed by a Turing machine as there are real numbers. Further, I noticed that the description or the definition of a real number is also finite (or at least can be, I haven't gone down the mental rabbit hole of figuring out what it means to have a non-finite definition for something). This means you can Gödel number them, order them, and then count all the numbers that can ever be described or defined with finite means, which then means there are again "that many" real numbers that can't ever be defined (which also means "most" real numbers can't ever be thought of or known to exist individually).
I was thinking whether there are numbers that can't be computed, but can be described and I found one. Order all Turing machines by their Gödel number and assign a natural number to each of them. Define the number h ∈ [0, 1] as having a 1 as its i-th decimal/binary digit if the i-th Turing machine halts and 0 otherwise. Then h is a unique real number, but evidently non-computable, since the halting problem cannot be decided.
I'd like to know whether I've made any mistakes anywhere in my reasoning. Otherwise I'd like to know if such a set of numbers already has a name, what other numbers or sets of numbers does it contain that are not computable, if this bears any significance at all or if it makes any sense to define this set and what other sets can be naturally thought of as being steps following rational, transcendental and computable number on the way to the reals.
Any improvements on this post are welcome.
