"Deal with" in detail, absolutely not, just as you say. Pardon my rephrasing you point (I think): You can show that some functions that give real numbers are c

"Deal with" in detail, absolutely not, just as you say. Pardon my rephrasing you point (I think): You can show that some functions that give real numbers are computable. (Let's say, computing that number digit by digit. Binary digit or base-ten, you choose.) You can't show that all Turing machines halt, so you can't use the mere existence of a Turing machine as proof of a computable function. Therefore, we can't know or compute the set of all real numbers because we can't compute the set of irrational numbers; there will always be some Turing machine that, unbenownst to us, calculates a (likely irrational) function that spits out a real number. If we can't describe even a way to sort functions into two piles: "computes a real number (digit by digit)" and "doesn't compute a real"; then we can't be said to "deal with" the set of real numbers at all well. This rather presupposes the (now pretty much accepted) concept that numbers "exist" if they can be computed (whether they have been computed, or conceived of, or not.) "Intuitionism" in other words, IIRC.