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). 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.
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). 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.