The number of points on a line is uncountably infinite. The number of lines on a plane is uncountably infinite. It seems like it follows that there would be an uncountably infinite number of points on a plane, too.
But it seems unsatisfying to believe that these are both the same thing. Surely adding an entirely new dimension must in some way increase the cardinality of what we're talking about, right? Or if not, is there a convincing demonstration that it doesn't change anything?
If they aren't, is there a way to show that they aren't? Are there mathematical ways of describing the distinction between the sizes of these sets, and if so, what are they called? (I'm having a lot of trouble searching for an answer because I'm not sure what words to use.)
Best Answer
To prove that $\mathbb{R}$ and $\mathbb{R^2}$ both have same size, it's sufficient to show that there is a bijection between these two. consider $f : \mathbb{R} \rightarrow \mathbb{R^2}$ which images each $x\in \mathbb{R}$ to $(x,0)$. this function is clearly one to one.
Assume another function $g : \mathbb{R^2} \rightarrow \mathbb{R}$.The function formula is this:
give any $(x,y) \in \mathbb{R^2}$ . Write x and y by their decimal expansion, so $(x,y)=(A_0A_1...A_n.a_0a_1...., B_0B_1...B_m.b_0b_1...)$ without loss of generalitty assume that $m\ < n$ . say $g(x,y)=A_0B_0A_1b_1...A_mB_mA_{m+1}...A_n.KFa_0b_0a_1b_1....$, which $K=0$ if x is positive and $K=1$ if x is not. and Also $F=0$ if y is positive and $ F=1$ if y is not. it's obvious that the function $g$ is one to one.
so by using the Schroeder-Bernstein Theorem there is a bijection between $\mathbb{R}$ and $\mathbb{R^2}$.
So, we proved that $\mathbb{R} \sim \mathbb{R^2}$ . Hence the number of points on a line is equivalent to number of points on a plane.