Cardinality of a line and a half plane intuitively it seems like the cardinality of the set of points that make up a line should be different than the cardinality of the set of points that make up a half plane but I couldn't come up with a proof, does a simple proof exist?
 A: It is easiest to start with showing that the unit interval and unit square have the same cardinality.  Ignoring the issue of decimals that end in an infinite series of $9$s (which can be patched up), take $x \in (0,1)$ and $(y,z) \in (0,1) \times (0,1)$  Construct a bijecton by taking all the odd place digits of $x$ to make $y$ and the even place digits of $x$ to make $z$.  So $0.123456789\dots \Longleftrightarrow (0.13579\dots, 0.2468\dots)$Now use your favorite bijection between the interval and the (half) line three times and you are done.
A: I'm pretty sure there isn't a proof that shows the cardinality of a line is different than the cardinality of the set of points in the half plane. Are you familiar with space filling curves?
http://en.wikipedia.org/wiki/Space-filling_curve#Outline_of_the_construction_of_a_space-filling_curve
I believe this result can be extended to show that the two cardinalities you mention are equal. Intuition means nothing when it comes to infinity, so don't get too hung up on things that don't seem right when infinity gets involved. Instead, appreciate its fascinating, mind-bending nature :)
A: A general answer
$$|\mathbb{R}^n|= |\mathbb{R}| = \mathfrak{c}$$
That is, any coordinate space in the real numbers has the same cardinality as the real numbers alone: the cardinality of the continuum.
So it is clear that a line is bijective to $\mathbb{R}$ and the halfplane is also bijective to $\mathbb{R}^2$, consequently, they have the same cardinality, the cardinality of the continuum $\mathfrak{c}$.
