I'm not a mathematician, but constructivists aim to define mathematics without uncomputable numbers, see

https://en.wikipedia.org/wiki/Computable_analysis

and

https://en.wikipedia.org/wiki/Computable_number#Use_in_place...

As far as I can understand, the set of all computable numbers (including all algebraic numbers and many transcendental numbers, such as Pi), even has the same cardinality as the rationals, and thus the natural numbers.

The reason we consider uncomputable numbers "numbers" include some definitions about infinite series and analysis that would need to have stricter requirements for convergence when looking only at the computable numbers, not the real numbers.

And defining a concrete bijection between the natural numbers and the computable numbers would also solve the halting problem and is impossible, we only know that such a bijection exists: defining it would mean to have an algorithm that can prove for a specific Turing machine that it is the minimal one computing it's output, among a given set of universal Turing machines / UTM encoding.

(please take this with a grain of salt as I'm stepping outside the bounds of my knowledge here)

Any given computation either halts or it doesn't. You can encode that information in a single bit, as a specific number. Since there is a countably infinite number of possible computations, you'd need a countably infinite number of bits.

So you can never find enough storage to hold the full solution of the halting problem in the real world. But you can find enough storage in a real number. Because real numbers can have a countably infinite number of digits after the decimal point. So you can stuff your countably infinite number of bits representing the solution of the halting problem in there.

Which specific real number you get depends on the details of the encoding, but it's definitely some real number. And it cannot be computed, because if it could, you could read the solution to the halting problem off its digits, but the halting problem is known to be uncomputable.

Here’s a nice concrete construction. To start, fix some enumeration ϕ of Turing machines. Let’s define a sequence of rational numbers x_k as $\sum_{i=0}^k 2^{-(i+1)} * halts(ϕ(i),k)$, where $halts(M,k)$ returns 1 if the machine M halts before taking k steps when fed the empty tape, and 0 otherwise. This is perfectly computable, as we only ever need to run a finite number of machines a finite number of steps for each k.

This sequence of rationals is monotonic and is upper-bounded by 1, but does not have a computable least upper bound. If such an upper bound existed, then it would encode solutions to the halting problem for every program. However, the reals have least upper bounds of all upper bounded subsets under mild classical assumptions, so we’ve made ourselves an uncomputable real out of computable data.

Sequences of this form are called Specker sequences, and are how you cook up most uncomputable numbers. There are models of constructive logic that do not admit any Specker sequences and admit only computable reals, but that is beyond the scope of a single comment :)

Enumerate all well formed programs in order. For programs that halt assign the digit 0, and for the ones that don't, the digit 1. Put the digits after a decimal point and interpret in binary.

I assume this refers to Chaitin's constant: https://en.wikipedia.org/wiki/Chaitin%27s_constant

Busy beavers are a classic example. They're mostly-hypothetical numbers that tell you "if any Turing machine of size s runs for longer than this, it doesn't halt." There's a link to that in the sentence you quoted.