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If we are limited by what we can construct with compass and straight edge,
then what are the fundamental curves required for constructing any real number?
In other words, what is the smallest collection of geometric objects for constructing a line segment of any length? And does that collection also allow for the construction of any angle?

This question is not limited to tangible tools. Any software tool or theoretical tool that is based on already constructed points, lines and circles is fine.

Honest Abe
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1 Answers1

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It seems the following.

If on each step of the construction we have a finite number of choices of operations, if we are based on already constructed points, lines and circles, there we can obtain only countably many results. So to construct a segment of each length and an arc of each angle we have to have a $\frak c$ many choices. But in this case we may simply enumerate all angles and real numbers as our tools.

PS. But the question with algebraic numbers instead of real seems not so trivial, so you may ask it as a different question at MSE.

Alex Ravsky
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