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Which is a technical way of saying "in real life people use finite rational or algebraic approximations for reals, so uncountability of reals and infinite preci

• eru 6y ago
  Not quite. Consider the following real number made of binary digits: Enumerate all Turing machines and all possible inputs, iff the i-th machine/input combination holds, the i-th binary digit in our number is 0, otherwise 1. This number is well-defined (once you fix your enumeration scheme). But there's no finite algorithm to produce approximations in your sense. What I was after were what's also called Computable numbers (https://en.wikipedia.org/wiki/Computable_number https://en.wikipedia.org/wiki/Computable_number). But I used the more general term of co-recursion, that also applies to arbitrary other data-structures like infinite lists, or with some generalization, infinite event-loops where the important condition is that each run through the body of the loop only takes finite time.