Construct(with ruler and compass) $\mathbb{R}^2$ from a non numerable set of points I just worked on fields and ruler and compass constructions in my college class. We did all constructions starting from points $(0,0)$ and $(1,0)$, but I feel that it's also interesting to think on what can be built if we where to start from a different set of points.
I came up with the following question I can't seem to solve. Can any point from $\mathbb{R}^2$ be constructed if we start from a non countable set of points from $\mathbb{R}^2$? I first thought that that wasn't true, but I can't seem to find a counter example. I feel like if I where to prove I'd try to use the existence of accumulation points, but I don't know what to do with that. Could anyone help me solve this problem or help me find a counter example?

Starting from an uncountable set may be not sufficient: think of an uncountable subfield $F\subset\Bbb C$ such that $\Bbb C$ is not algebraic over $F.$ For instance, let $B$ be an transcendance basis of $\Bbb C$ over $\Bbb Q,$ $b\in B,$ and $F=\Bbb Q(B\setminus\{b\}).$
