5 ms·
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-
by ghayes 8y ago
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).
- alew1 8y agoWe 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.
- OscarCunningham 8y agoBoth representations have the feature that it's impossible to check whether two real numbers are equal.
- goldenkey 8y agoActually with the physical objects you could use one to occlude the other and test for light passage
- empath75 8y agoNot with infinite precision.
- aeneasmackenzie 8y agoImportantly, this is actually true of the real numbers.
- OscarCunningham 8y agoI don't think I believe in "actual" real numbers.
- dllthomas 8y agoThat can't represent most real numbers, although I don't know that we have actually established that physical lengths can.
- iitalics 8y agoThe 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.