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In some sense the ruler-and-compass constructions are even more computable than the natural numbers: Any (first-order) statement in Euclidean geometry is either
by remcob 8y ago
In some sense the ruler-and-compass constructions are even more computable than the natural numbers: Any (first-order) statement in Euclidean geometry is either true or false and there is an algorithm (on a Turing machine) that can compute it. Natural numbers don't have this property due to Gödel incompleteness.
In practice, anything you do with a ruler and compass has an equivalent representation in polynomials that is exactly computable.
- kps 8y agoFor the non-math readers, ruler-and-compass constructions are constructible numbers¹, which are a subset of algebraic numbers², which are a subset of computable numbers³, which are a tiny⁴ subset of complex numbers. ¹ http://mathworld.wolfram.com/ConstructibleNumber.html http://mathworld.wolfram.com/ConstructibleNumber.html ² http://mathworld.wolfram.com/AlgebraicNumber.html http://mathworld.wolfram.com/AlgebraicNumber.html ³ http://mathworld.wolfram.com/ComputableNumber.html http://mathworld.wolfram.com/ComputableNumber.html ⁴ http://mathworld.wolfram.com/MeasureZero.html http://mathworld.wolfram.com/MeasureZero.html