On the contrary, we do have a choice! We could use computable numbers instead of reals. A computable number is any number which is output by some Turing machine, or, equivalently, any number which can be found by some algorithm. e and pi are computable. You are right that Riemann integrals won't exist, but if you modify definitions somewhat, derivatives and integrals can be defined just as easily for computable numbers as for reals and you can do calculus with them (see second link).
Unlike reals, computable numbers are countable, and are all describable. (That is, there are aleph-null of them, so there are exactly as many computable numbers as natural numbers, and fewer than real numbers). And while almost all real numbers aren't computable (by the argument in the article), essentially every number you'd ever stumble upon in a math class is.
I prefer computable numbers to reals because I have trouble accepting that a thing exists when it by principal cannot be defined.
Unlike reals, computable numbers are countable, and are all describable. (That is, there are aleph-null of them, so there are exactly as many computable numbers as natural numbers, and fewer than real numbers). And while almost all real numbers aren't computable (by the argument in the article), essentially every number you'd ever stumble upon in a math class is.
I prefer computable numbers to reals because I have trouble accepting that a thing exists when it by principal cannot be defined.
https://en.wikipedia.org/wiki/Computable_number
http://www.amazon.com/Computable-Calculus-Oliver-Aberth/dp/0...