The Axiom of Choice: If $a$ is a class of non-empty sets $x,$ there exists a function $f$ such that $f\left(x\right)\in{x}$ for all $x\in{a}.$
https://mitpress.mit.edu/contributors/h-behnke
The reason this question is relevant to the axiom of choice is that the axiom provides us with a function that chooses an element form any set of real numbers, but the function does not specify the chosen element.
I have added the adjective nominated as a synonym for specified. That is, specifying may also be considered naming. I have given an example of a named real number which is not an element of the countably infinite set of rational numbers. It is asserted that there are uncountably meany such real numbers.
My understanding of the theory of formal systems tells me that, since any formal system must be specified using some necessarily finite vocabulary, there are at most a countably infinite number of producible instructions. Since an algorithm is a set of instructions, there exist at most a countably infinite number of algorithms of any formal system.
It appears to me that any specified real number must correspond with at least one algorithm for accurately determining, to an arbitrary precision, its value relative to the rational numbers. For example, the specific number $\pi$ can be approximated by a number of different methods, all of which begin with an initial condition stated in a finite number of symbols. The algorithm for producing each subsequent refinement of the current approximation must also be stated in a finite number of symbols.
We thereby specify a real number, the value of which cannot be precisely determined in a finite number of steps, nor can its value be given in "closed form" using a finite number of symbols.
So, even though we may argue that there is an uncountable infinite number of real numbers, we can only specify at most a countably infinite subset of real numbers.
I will call those real numbers which are given by an approximating algorithm specific real numbers. Real numbers which are not specifically identified by this means I will call anonymous real numbers.
Assuming that it has been proposed, where has this distinction between specific and anonymous been proposed before?
"The Tao that can be told is not the eternal Tao" ~~ Laozi
The notion you are calling “specific real number”, which you are using to mean a number whose digits can be computed by an algorithm, is a notion in mathematics. The standard term for it is computable real number. And as you said, such numbers are countable, because there are only countably many algorithms.
However, computable real numbers do not even exhaust the real numbers which can be unambiguously specified. That’s because it’s possible to be unambiguously specify an real whose digits cannot be computed by any algorithm, for instance Chaitin’s constant. So there is a broader notion of real numbers which can be unambiguously specified. Those are called definable real numbers, and they too are countable. Because there are only countably many definitions.