I have been studying point constructions from ruler and compass in context to field extensions. I follow the proofs without too much trouble, but there is a simple note I do not follow.
In A Book of Basic Abstract Algebra by Charles C. Pinter a point $P$ is defined to be constructible from a set of points $A$ in a plane if there are points $P_1 , P_2, ... , P_i = P$ such that $P_1$ is constructible in one step from $A$ and $P_2$ is constructible in one step from $A \cup P_1$ and so on til $P_i$ is constructible in one step from $A \cup \{P_1 , ... , P _i \}$
I was hoping someone could help me explain how we can define one point to be several points? It is: $ \space $ $P_1 , P_2, ... , P_i = P$ $ \space $ I am referring to.
In advance; thank you :D