Finding a point in an infinite space

[JCarter426]

I don't agree, because even an arbitrarily precise approximation isn't the actual number. Getting the exact irrational number would require infinite time, which I think doesn't match the definition of "computable".

The definition of computable:

A number which can be computed to any number of digits desired by a Turing machine.

That means a finite number of digits is fine.

The issue isn't an infinite amount of digits - not all irrational numbers are uncomputable, and actually all the ones we know are computable. The issue is whether a finite process can determine what those digits are.

A number like π is irrational and transcendental but still computable. We can take the ratio of a circle's circumference to its diameter to determine the value to whatever degree of precision we want. That's a finite amount of steps that can give us the number to whatever decimal place we want. In that sense it is computable.

An uncomputable number is one for which no such process exists. You'd truly need an infinite amount of steps to determine the digits. Which is exactly what an unlimited series of yes/no guesses gives you, provided that somebody magically knows the answer already.

Which is another way of saying this scenario could never exist.

Which, I mean: that's obvious, but it's kinda neat stumbling over an actual proof for it.

Indeed.