How many paths exist between two points in the plane? Fix distinct $a,b \in \mathbb{R}^2$. In terms of cardinality (say, beth numbers), how many distinct continuous functions $f : [0,1] \rightarrow \mathbb{R}^2$ satisfying $f(0)=a, f(1)=b$ are there? (Presumably, the answer is independent of which $a,b$ are chosen.)
It seems clear that the answer is either $\beth_1$ or $\beth_2$.

Clearly there are at least $\beth_1$ paths: for any $x \in (0,1)$, consider the path that sits in $a$ until time $x$ and then heads towards $b$ at constant speed. For the upper bound, note that a continuous function from $[0,1]$ to $\Bbb{R}^2$ is determined by its values at rational points, so there are at most $(\beth_1^2)^{\aleph_0}=\beth_1$ such functions.
