Even assuming AC, how could one ever "choose" a unique, non-algorithmically specifiable, [like $\pi$, $e$, & $\varphi$ are] transcendental real number from, say, I?
This seems like a fairly simple question, but it seems to me to border on some important related topics in Real Analysis and Set Theory. The problem is that, apart from a finite, or at most a countable, set of transcendental numbers that can be approximated, but hardly fully specified, by a decimal string [assuming base $10$] produced by some algorithm, the other, infinitely more numerous, transcendental real numbers in any interval cannot be indicated by any algorithm.
Worse still, there would seem to be no way even to establish that such random, run-of-the-mill, transcendentals exist at all since any specification we might attempt to give for one needs to break off at some finite point in the decimal enumeration meaning that either $1$.] all of the uncountably many possible further digital enumerations, because they remain inherently indiscernible from one another, -- at least to us! -- and thus fall under Leibniz's principle of the "identity of indiscernibles," must constitute but ONE transcendental number, or, -- considering that such a first alternative is hardly very appealing, $2$.] there really does exist some vast uncountable set of possible further continuations of the string of digits which, if it could be carried out fully, perhaps by some angel or goddess, would be able eventually to specify uniquely each transcendental number beyond the point of human exhaustion, -- another highly metaphysical claim which strikes me, at least, as being ostensibly dubious.
Thus it would seem that we may have answered my initial question here by finding that in fact one can NEVER"choose" a unique, non-algorithmically specifiable, transcendental real number from a real interval. One can instead merely pluck out a huge set of uncountably many such numbers, each as close to the terminated string of digits as the next digit in the sequence would be,or one must find some other way to specify which one of all those "candidate" numbers is the single one you mean to choose, assuming that such a thing could ever be possible.
All of these concerns would be but a humorous exercise in metaphysics turned sour, of course, except that, at least if I remember correctly, there are several proofs in both Set Theory [and Real Analysis too] which depend upon one's being able to specify a unique, and often even "random," transcendental number. The most obvious of these would probably be the proof, if such there be, that, as indicated earlier here, the probability of choosing a single real from the unit interval, I, that is not also a transcendental is exactly O, -- since even all the algebraics together are still a countable set of measure $0$. Clearly such a proof, and many others like it, are already making several assumptions about the set of transcendentals that, if the analysis above is at all correct, seem highly dubious, if not outright specious.