Quote Originally Posted by crayzz View Post
I think this is a misunderstanding of infinity. Yeah, we can describe numbers, even irrational and transcendental ones, through countably infinite steps, but you won't actually reach the target. If it takes countably infinity steps, by definition you do not get to reach the target.

Think of it this way: a lot of the methods suggested can only ever produce a rational number; some are worse and can only produce a decimal/binary number. Such a method could never reach an irrational or transcendental number. It could get close, as close as you want. But they necessarily can not reach the target.

EDIT:

We can reach the target in the same sense that we can calculate pi or describe 1/3 in decimal notation, which is to say we can't.
Considering that the proposed problem allows infinite amount of questions as a valid solution then we actually do reach the sought number in the limit. Being infinitely close is the same as reaching the target as for example 1=0.9999999... by the very definition of real numbers. It would stop being true if we extended real numbers with infinitesimals, but it is not the case.

Besides, the part you quoted concerns finite precision search, which boils down to searching for a pair of intigers and those can be always found with finite amount of questions, since they are countable.