Do I see it right that a randomly chosen number $a$ of the form $ a = 0.abc...xyz$ with random digits $a,b,c,x,y,z$ an approximation of a transcendental real number is?
It is an approximation of a transcendental number to any finite precision, I assume.
Now, is it approximating a transcendental number for sure?
I think it is, because there are infinitely more transcendental numbers than algebraic numbers. So the chosen number is either algebraic, or between two algebraic numbers. As there are infinite numbers between, the number is one of them.
With probability $ p = 1 $, and not in any way $ p < 1 $.
That would mean $a$ is transcendental for any practical purpose $u$.
And $a$ is transcendental for any purpose $u$, for any finite definition of $u$, which may not be practical.
It is even transcendental for anything that is not a purpose. Any definition of $u$ is finite. Because the visible universe is finite.
Now, is that right?
(Obviously, $a$ is not strictly transcendental, because it is finite. But in which sense is it not transcendental - it is chosen, and the chosen number can be any number whatsoever. A chosen number must be encoded, and it would be nice if all mass-energy of the earth, excluding us, is enough. Or say our galaxy, and Andromeda, to be generous. But feel free to encode with the visible universe. Without us.)