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Comment by ghayes | Hacker News Reader

Doesn't computable numbers, from the reference Wikipedia article, only state they can be computed to a given nth digit? Thus, in the OP's example, the rule-and-compass construction is exact whereas the Turing machine could only approximate the number (unless given infinite time, which I assume is out).

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We need a digital representation of real numbers. One possible choice is: a real number r is represented as the source code of a function which takes a natural number n to the nth digit of r. This seems no more like “cheating” (and to me, it is less like cheating) than to represent a real number r by a line segment of length r.

Both representations have the feature that it's impossible to check whether two real numbers are equal.

Actually with the physical objects you could use one to occlude the other and test for light passage

Not with infinite precision.

Importantly, this is actually true of the real numbers.

I don't think I believe in "actual" real numbers.

That can't represent most real numbers, although I don't know that we have actually established that physical lengths can.

The catch is that you can't actually measure the rule-and-compass construction with infinite precision. You can perform more operations on the construction and then measure later to verify that it produced the correct result, but so can a turing machine acting on a mere "representation" of the number.

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