Still at some of my earliest encounters with the elementary concepts of Set Theory, I understand what the Power set of a set is but I have a hard time wrapping my head around what the definition implies in the case of a set like $\Bbb R^2$.
I understand that any function $f:\Bbb R \rightarrow \Bbb R$ is a subset of $\Bbb R^2$
$$\lbrace (x,f(x)): x\in\Bbb R\rbrace \subseteq \Bbb R^2$$
and I can also see why this extends to any set of points in the Cartesian plane
$$\lbrace (x,y): x\in\Bbb R \wedge y \in\Bbb R\rbrace \subseteq \Bbb R^2$$
but then, when I think about what this could mean, my brain starts to hurt a bit. If I can think about something and then write that something down or make a sketch of it, it becomes a collection of points in a plane ergo, by $\lbrace (x,y): x\in\Bbb R \wedge y \in\Bbb R\rbrace \subseteq \Bbb R^2$, it is an element of $\mathscr P(\Bbb R^2)$.
Does that imply any drawing, text, plan, map or even this post could be considered a subset of $\Bbb R^2$ such that $\mathscr P(\Bbb R^2)$ contains any $2$-dimensional "object"?
Since I can make a sketch of most of the things I can think about, does that mean $\mathscr P(\Bbb R^2)$ contains anything that I could think about and sketch and even anything I could sketch but will never think about as well?
Thanks for taking the time to provide your input, I appreciate it a lot!