Programs written in programming languages aren't guaranteed to terminate. Computable numbers, however, are all real numbers that can be represented to arbitrary precision by a computation that terminates. Wouldn't that mean we don't need a fully fledged programming language to have the capability to represent all computable numbers?
Somewhere in there is the idea we just need a program that determines whether an arbitrary program terminates or not...
I can't tell how much of this is a joke, but, yes, it turns out you can't write such a programming language for exactly the reason you allude to: you can't tell whether a given number is computable in finite time, as you don't know whether your program will terminate.
Programs written in programming languages aren't guaranteed to terminate. Computable numbers, however, are all real numbers that can be represented to arbitrary precision by a computation that terminates. Wouldn't that mean we don't need a fully fledged programming language to have the capability to represent all computable numbers?
Somewhere in there is the idea we just need a program that determines whether an arbitrary program terminates or not...