Axiom of choice and Euclid's first proposition

Question

Euclid's first proposition constructs an equilateral triangle given a segment $AB$ via two circles using the segment as radiuses, using $C$ as one of the intersection points of the circles.

I understand that there assumptions made which are not part of the original set of axioms. E.g. that the circles intersect (i.e. such a $C$ exists). Additional axioms would be required e.g. involving some continuity axiom for intersection issue.

But when it turn out that the circles intersect twice, by what criteria is one of the two points chosen? Does this require something similar/equivalent to the axiom of choice or not?

Either choice yields an equilateral triangle, and I don't believe any claim is being made that requires a distinction between the two possibilities. — Greg Martin, Mar 02 '21 at 23:57
And in any case the axiom of choice is not needed to choose a single element from a non-empty set. — Brian M. Scott, Mar 03 '21 at 00:04
I agree it's a duplicate, it can be deleted, does not add more on top of the existing one — Mircea Baja, Mar 04 '21 at 09:03

score 1 · Accepted Answer · answered Mar 03 '21 at 00:05

1

No, the axiom of choice is not needed to provide an element of a single finite set.

What the axiom of choice says is that the product of a set of nonempty sets is itself nonempty.

answered Mar 03 '21 at 00:05

NoLongerBreathedIn

222

Axiom of choice and Euclid's first proposition

1 Answers1