Finding a point in an infinite space

[crayzz]

These methods are not actually less powerful. The method that could determine a binary representation of the number, could determine a binary representation of anything. For instance, the binary encoding of any rational number, assuming there exists a way to uniquely encode rational numbers in binary, which there does.

I think there is a misunderstanding with regards to countable infinity. Just because there are countably infinite steps does not mean that it is impossible, because time doesn't matter in this hypothetical scenario. Each step might be evaluated on a magical device so that it takes no time. Then we would arrive at an answer. The answer would have infinite information, but it would still be an answer. We can represent 1/3 in decimal notation, it is a 0.3 followed by infinite 3s. Just because actually writing it out would be impossible does not mean it doesn't exist, and neither does it not exist because there are infinite digits (side note, 2 has infinite digits in its decimal representation. It just so happens that most of them are 0). Numbers are not a process.

Even if the algorithm did not take 0 seconds, it could still be completed in finite time. What if each successive step was executed twice as fast, and the first took 1 minute. Then the entire process would take 2 minutes.

The point is, with a countably number of steps and an omniscient oracle, you can determine exactly a number so long as the steps take a finite amount of time in sum.

—Caerulea

I shouldn't have said we can't describe 1/3 in decimal notation, since we have an actual notation for repeating decimals. But it is true that a algorithm that can only produce decimal numbers will never spit out 1/3.

And that's exactly the point I was trying to argue against! If your algorithm can only produce a certain type of number, you can't get outside that type of number no matter what. Proposing countably infinite steps in finite time sounds nice, but I don't think it actually works. It violates what we know about computation for one thing. A turing machine of a given complexity will either halt or loop forever: you don't get to have it halt after infinite steps.

This, incidentally, brings up a practical concern with actually implementing the algorithm even in the integer case: without knowing how big the number actually is, we can't know that we've chosen a turing machine "large" enough to handle it. But setting that aside, no turing machine could ever compute a transcendental number to exact precision, or an uncomputable number to even arbitary precision.