From what I have read, all finite sets are countable but not all countable sets are finite. As I understand it,
- Countably Finite — a one to one map onto $\Bbb{N}$ with a limited number of members
- Countably Infinite — a one to one map onto $\Bbb{N}$ with an unlimited number of members but that you can count in principle if given an infinite amount of time
- Uncountably Infinite — there is no one to one mapping onto $\Bbb{N}$. Even if you count, you will miss some of the members. And it is infinite.
From this I gather that countable is not the same as finite. Countable is the one to one property with $\Bbb{N}$. Finite just means a limited number of elements.
Now consider $[0,1]$ which is closed and bounded.
- Bounded — $\forall k \in [0,1]$ we have $k \leq 1$. Similarly all $k\geq0$.
- Closed — it contains the endpoints $0$ and $1$
Yet I read $[0,1]$ is uncountably infinite. So clearly, neither closure nor boundedness implies finiteness or countability.
Question:
Why can a closed, bounded interval be uncountable?
It just seems like something that is bounded would be "more finite" than something that isn't.
Best Answer
I suspect you're conflating two meanings of "finite". Some sets are finite, meaning they have only finitely many elements. An interval like $[0,1]$ is not such a set. On the other hand, $[0,1]$ has finite length, which is a quite different matter. As the other answers have explained, finite length does not imply finiteness (or even countability) in terms of the number of elements.